
Book, ' CfcT 



Copighl X° 



COPYRIGHT DEPOSIT. 



ELEMENTS 



OF THE 



INFINITESIMAL CALCULUS 



BY 



G. H. CHANDLEK, M.A. 

Professor of Applied Mathematics, McGill University \ Montreal 



TRIED EDITION, REWRITTEN 
FIRST THOUSAND 



. <5 

- 



NEW YORK 

JOHN WILEY & SONS 
London : CHAPMAN & HALL, Limited 

1907 






LIBRARY of CONGRESS 

Two Copies Received 

JAN 7 1907 

Copyright Entry 

LASS J\ XXc„No. 

COPY B. 



■ U MII I lull 



Copyright, 1907 

BY 

G. H. CHANDLER 



ROBERT DRUMMOND, PRINTER, NEW YORK. 



PREFACE. 



The following pages are intended to serve as an introductory 
manual of Infinitesimal Calculus for beginners generally, but 
more especially for students of Engineering and other branches 
of Applied Science. 

As a logical foundation of the Infinitesimal Calculus the 
doctrine of Limits must be accepted as essential, but an 
attempt has been made at an early stage to accustom the 
reader to those principles and operations which are used in 
the practical applications of the subject. 

The order in which the subject is developed, though differ- 
ing from that of many text-books, is believed to be well calcu- 
lated to meet the difficulties and secure the interest of the 
student. 

In the present edition many changes and some additions 
have been made which it is hoped will bring the book into 
accord with present-day treatment and needs. 

To supplement the ordinary Mathematical Tables I have 
added short tables which are intended to facilitate curve 
tracing as well as the rapid calculation of integrals, etc. 

My thanks are due to Professor Murray Macneill for sug- 
gestions and for assistance in proof-reading and in the verifi- 
cation of examples. 

G. H. Chandler. 
Montreal, December, 19C6. 

iii 



CONTENTS. 



CHAPTER PAGE 

I. Limits. Infinitesimals 1 

II. Functions. Derivatives. Differentials . . .11 

III. Differential of a Power, a Product, and a 

Quotient . ■ 16 

IV. Tangents and Normals 20 

V. Differentials of Exponentials and Logarithms, 25 

VI. Differentials of Direct Circular Functions. . 28 

VII. Differentials of Inverse Circular Functions . 32 

VIII. Differentials of Hyperbolic Functions ... 34 

IX. Differentials as Infinitesimals 37 

X. Functions of more than one Variable ... 41 

XI. Small Differences . . . 47 

XII. Multiple Points 51 

XIII. Asymptotes 57 

XIV. Tangent Planes. Tangents to Curves in Space 62 
XV. Successive Differentiation 68 

XVI. Rates 72 

XVII. Maxima and Minima 75 

XVIII. Curvature 82 

XIX. Integration. Elementary Illustrations . . 93 

XX. Fundamental Integrals I .98 

XXI. Fundamental Integrals II 102 

XXII. Fundamental Integrals III 105 

XXIII. Fundamental Integrals IV 109 

XXIV. Integration of Rational Fractions . . . .112 
XXV. Integration by Substitution 114 

XXVI. Integration by Parts 120 

XXVII. Successive Reduction 124 

XXVIII. Certain Definite Integrals . . , 5 . .129 

v 



VI 



CONTENTS. 



CHAPTER PAGE 

XXIX. Areas and Lengths of Plane Curves. Surfaces 

and Volumes of Solids of Revolution . . .134 
XXX. Simpson's Rule. Volumes from Parallel Sec- 
tions. The Prismoidal Formula. Length of 

a Curve in Space 142 

XXXI. Polar Coordinates . .149 

XXXII. Associated Curves . . . ... . . . .165 

XXXIII. Centres of Gravity 176 

XXXIV. Moments of Inertia 181 

XXXV. Successive Integration 189 

XXXVL Mean Values 202 

XXXVII. Intrinsic Equation of a Curve. The Tractrix. 

The Catenary 205 

XXXVIII. Infinite Series 211 

XXXIX. Taylor's Theorem 223 

XL. Fourier's Series 228 

XLI. Approximate Integration. Elliptic Integrals 239 

XLII. Singular Forms 244 

XLIII. Successive Differentials of Functions of more 
than one Variable. Extension of Taylor's 
Theorem. Maxima and Minima from Taylor's 

Theorem 250 

XLIV. Differential Equations of the First Order . 259 
XLV. Differential Equations of the Second Order. 272 

Appendix. Note A. Partial Fractions 287 

Note B. Curve Tracing 290 

Note C. Hyperbolic Functions 294 

Note D. Mechanical Integration .... 299 

Miscellaneous Examples 305 

Tables. 1. Powers, Napierian Logarithms, etc. . . . 310 
' 2. Circular Functions I 312 

3. Circular Functions II 313 

4. Hyperbolic Functions 314 

5. Lambda Function 315 

6. Gamma Function 315 

7. First Elliptic Integral 316 

8. Second Elliptic Integral 316 

Index 317 



ELEMENTS 



OF THE 



INFINITESIMAL CALCULUS. 



CHAPTER I. 
LIMITS. INFINITESIMALS. 

i. Constant. Variable. When a quantity remains un- 
changed while another quantity changes, the former is called 
a constant, the latter a variable. 

2. Limit of a variable. If the value of a variable v ap- 
proaches nearer and nearer to that of a constant a in such 
a way that their difference becomes and remains less in 
absolute * value than any given positive number, however 
small, v is said to approach a as a limit, and a is called the 
limit of v. 

If a; = 2,f the definition implies that the absolute value 
of x — 2 becomes and remains less than any positive number 
we choose to assign; e.g., it becomes <10 -3 ; it further 

* The absolute (or arithmetical or numerical) value is the value 
without regard to sign. Parallel vertical lines are used to indicate 
an absolute relation; thus |— 2| =2 expresses that the absolute value 
of — 2 is 2. So also x|<|a| or :r|<|a indicates that the absolute 
value of x is less than that of a . 

f The symbol = signifies an approach to a limit. Thus x = 2 may 
be read: x approaches 2 (as a limit). If the words "as a limit " are 
not expressed, they must be always understood. 



2 INFINITESIMAL CALCULUS. [Ch. I. 

diminishes, becoming less than any smaller given positive 
number (say 10 -6 ), and so on. While x = 2. x—2 cannot 
be zero: the definition does not imply that x acquires the 
value 2. If it does become 2. it is no longer approaching 2 
as a limit. 

Ex. 1. The value of (j : -4 x-2) is equal to that of x + 2 
when any number except 2 is substituted for x. When j*=2 the 
fraction takes the form 0. an expression which is undefined 
and meaningless, but when x = 2 the limit of the value of the 
fraction is equal to that of J*-r2, i.e., 4, or in symbols * 



<~£?) - 



Similarly, if y=2x-x : . - = =2. 

2. being the radian measure t of an acute angle, 6 lies be- 
tween sin and tan 0. Hence, dividing each of these into sin 0, 
sin 0/0 lies between 1 and cos 6. But cos = 1 when = 0. 
Hence 

/sin 6\ 






3. The sum of the terms of the series 1— i + J— i-K . . has a 
limit as the number of terms increases without bound. 



Fig. 1. 

Let s n stand for the suni of the first n terms. Take 0P 1 =1, 
P,P_=-±. PJ* a =h etc. Then s x =0P ly s 2 =0P z , s,=0P, y etc. 
The points with odd subscripts continue to move to the left, 



* The symbcl £ is used for •"the limit of" (Echols. Differential and 
Integral Calculi. 

- The radian | =&7°'2958 . . .=206265") will be always understood 
to be the unit angle unless the contrary is manifest. 



3. 4.] 



LIMITS. INFINITESIMALS. 



those with even subscripts to the right ; they tend toward meeting 
at some point P. since the fractions which are added = 0. Hei 
\OP — s n \ becomes and remains less than any given positive 
number, however small: therefore OP is the sum limit. The 
limit is that of the endless decimal '69314 . , . 

A series which has a limit is said to be convergent. 

4. If the series \— f+f — £+. . . be similarly represented, the 
two sets of points cannot come within a distance 1 of each 
other, since the absolute value of the fractions added =1. Sup- 
pose the even P's to approach a point P (really the point P of 
Ex. 3) as a limit of position. Then OP— s n \ may become less 
than any given positive number, however small, but it will not 
remain so, for the addition of a single term changes OP — : 
by an amount approximately equal to 1. Consequently there is 
no limit, or the series is non-convergent. (Observe the signifi- 
cance of the words "and remains'' in the definition of § 2). 

3. A variable quantity may or may not be capable of as- 
suming a value equal to that of the limit which it approach— 
Thus in Ex. 1. x — 2 = 4 when x = 2. and x — 2 = 4 when 
x = 2. Also (x 2 — 4) {x— 2) =4 when x = 2. but it cannot =4. 
for when x = 2 there is no fraction. Again, the series 
1 — J-fJ— . . . can never equal its limit. In certain cases the 
limit and the value are entirely different. For example 
(Ch. XL), sin x—\ sin 2x — \ sin 3.r— . . .=\~ or — \- ac- 
cording as x=n by increasing or decreasing, but =0 when 

rm mm 

wU /. . 

4, A variable may approach nearer and nearer to its limit 




in three ways: (1) by increasing only. (2) by decreasing 
only, (3) and by increasing and decreasing. Thus. Fig. 2 ; 



4 INFINITESIMAL CALCULUS. [Ch. I. 

when M moves to the right v may be supposed to approach 
the limit a if P moves along A B only, (2) along CD only, 
or (3) from AB to CD, back to AB, etc., its value continually 
approaching a. 

5. Limit of a constant. It is sometimes convenient to 
regard a constant as its own limit. Thus £ x=0 (ax + b) = b* 
and hence if a = 0, £ x ± b = b. So also £ Xz ± ax/x = a. 

6. Infinitesimals. An infinitesimal is a variable whose 
limit is zero. It is therefore a quantity which approaches 
nearer and nearer to in such a way that its absolute value 
becomes and remains less than any given positive number, 
however small. Thus x is infinitesimal when # = 0; if x 
actually becomes it is no longer infinitesimal. 

Ex. When x is infinitesimal the following are also infinitesimals: 
x 2 , sin x, tan x, 1 —cos x, log (1 +x), 1 —2 X . 

7. Quantities which are infinitesimal are such solely on 
account of their tending to a limit zero, not because of their 
having arrived at any particular degree of smallness; in other 
words, their chief characteristic is not being small but getting 
smaller. Nevertheless, it is often convenient and sometimes 
necessary to suppose them very small when they begin to 
= ; hence it is customary to regard them as very small 
in all cases. 

8. From the definition' of § 2 it follows that the difference 
between a variable v and its limit a is an infinitesimal. Hence 
v — a = i, or v = a+i, where i is infinitesimal. Also if v — a=i, 
or v = a + i, where v is a variable, a a constant, and i an 
infinitesimal, a is the limit of v. 

Notice that the sign of v is the same as that of a as soon 
as the absolute value of i becomes less than that of a. 

9. Infinites. A variable which is increasing (or decreasing) 
without bound, i.e., so as to exceed in absolute value any 

* As in algebra, letters near the beginning of the alphabet are used 
for constants unless the contrary is obvious. 



5-12.1 LIMITS. INFINITESIMALS. 5 

given positive number, however large, is called an infinite, 
and is represented by oo (or — oo ). Such a quantity has no 
limit which accords with the definition of § 2. 

It should be noticed that an infinite, like an infinitesimal, 
is a variable. If z = 0, l/z=oo or — oo, and if z=oo* or 
— oo, l/z = 0. 

Ex. When x = 2, x— 2 = 0, and l/(x — 2) = oo or — oo accord- 
ing as x approaches its limit by decreasing or increasing. 

When x = J^, tan x = oo or — oo according as x is increasing or 
decreasing. 

io. Quantities which are neither infinitesimal nor infinite 
are said to be finite. A finite variable is therefore one whose 
value stops short at some number which can be assigned. 

ii. If i is an infinitesimal and n any constant, ni is an 
infinitesimal. 

For |ra|< any assigned positive number a if i\<\a/n; 
but i does become | < \a/n f since a/n is also an assignable 
number. In special cases n may be 0, then ni remains =0. 

If n is infinite, ni may be infinite, or it may have a limit, 
which may or may not be zero. 

12, The sum of any finite number n of infinitesimals is 
an infinitesimal. 

Let i be a positive infinitesimal which is and remains 
greater than the absolute value of any of the given infini- 
tesimals. Then the sum of the given infinitesimals <ni 
and > —ni, and is therefore infinitesimal (§ 11). 

If n is infinite, the sum may have a finite limit. The deter- 
mination of such a limit is the fundamental problem of that 
part of the subject which is known as the Integral Calculus. 



Ex. 1+1+.. . +^ = I^±l ) =^(i + ^). The limit of the 
n 2 n 2 n 2 2 n 2 



sum for n infinite is therefore J. 



*z = co should be read "x is a positive infinite," or u x increases 
without bound"; x— —go, u x is a negative infinite," or "x decreases 
without bound," 



6 INFINITESIMAL CALCULUS Ch. L 

13. Propositions relating to limits. Let v 1; v 2 be two 

variables which have limits a x , a 2 . Then V\-=a,i+ii and 
V2 z =a2 + i2i where i\ and i 2 are infinitesimals. 

(A) The limit of the sum (or difference) of the variables 
is equal to the sum (or difference) of their limits. 

For, (vi +v 2 ) - (ai +a 2 ) = i\ +%& 

which is infinitesimal. Hence £(^1 + ^2) = 7 -i +&2- 

The proposition is evidently true for the sum of any finite 
number of variables. 

(B) The limit of the product of the variables is equal to 
the product of their limits. 

For, v\V 2 — a x a 2 = {a\ + i\ ) (a 2 + i 2 ) — a x a 2 

= a 2 i\+a\i 2 +i\i 2 > 

which is infinitesimal. Hence £v\V 2 = aia 2 . 

This also is true for any finite number of variables. Thus 
if x = a, £ 2 = a 2 , x 3 = a 3 , etc. 

(C) The limit of the quotient of the variables is equal to 
the quotient of their limits, provided that the limit of the 
divisor is not zero. 

For, ifa 2 ^0, ^-^ ai+k Ul a ^~ a ^ 



v 2 a 2 a 2 +i 2 a 2 a 2 2 +a 2 i 2 '' 

which is infinitesimal, since the numerator = 0, while the 

denominator = a 2 2 . Hence £— = — . 

^2 &2 

If a 2 = and ai^O, V\/v 2 is infinite and therefore has no 
limit. If a 2 = and ai = 0, V\/v 2 is the quotient of two 
infinitesimals and may have a limit, as in Exs. 1 and 2 of 
§ 2. The determination of such a limit is the fundamental 
problem of that part of the subject which is known as the 
Differential Calculus, 



13, 14.] LIMITS. INFINITESIMALS. 

sin 

■m i «. . „ sin tan 

Ex. 1. Since tan = 



cos 0' cos 



Hence (§2, Ex. 2), £ d ^(^^jJ- = l. 

/sin 0\ 2 

2. Since 1 -cos = — — , . . — =-— ; 

1 + cos 2 1 + cos 

• r A-cosfl \ 1 _1_ 

... sin 3 . . /tan0-sin0\ 1 

3. tan -sin 0= — — rr, . - £ e =o\' Zi ) = o* 

COS 0(1 + COS 0) o v\ Q3 / 2 

n-2~\ 2' " ^°°\n-2/ * 

a + b cosi(A— B) 



1 

n 




5. In any plane triangle, fl cosiU+jB) - 

If a and & are tangents at two points near * 
one another on a curve, and the points ap- 
proach coincidence, A and B = and there- 
fore the right-hand member of the above ^= 1 . 
Hence £(a + b)/c = l when c = 0. We may 
assume that the arc > c and <a + &. Hence in any curve the limit 
of arc/chord is 1 as arc and chord = 0. 

n o.i ,i i r 3x 2 —2x 3a 2 — 2a . . . _ 

6. know that £„. „t-^- ^ = r~^ n> provided that a is 

x=a 4x 3 + x — 2 4a 3 + a-2 

not a root of the equation 4# 3 + £ — 2 =0. 

14. Orders of infinitesimals. If the limit of /?/«, the 

quotient of two infinitesimals, is zero, /? is said to be of a 

higher order than a. If /?/a has a limit which is not zero, 

3 is said to be of the same order as a. If /?/a n (where n is a 

instant) has a limit which is not zero, /? is said to be of 

le nth order, a being assumed to be of the first order. 

* It is assumed that the points may be taken so near each other 
that the arc is everywhere concave to the chord. 



8 INFINITESIMAL CALCULUS. [Ch. I. 

Ex. 1. a and /? being of the first order, a/? is of the second 
order, a 2 /?, a/? 2 of the third, a 3 /?, a 2 /? 2 , a/? 3 of the fourth. 

2. If is infinitesimal and of the first order, sin 6 and tan d 
are of the first order, 1 —cos d of the second, tan 6— sin 6 of the 
third. 

15. Definition. When the limit of the quotient of two 
infinitesimals is 1 the infinitesimals are said to be equiva- 
lent.* 

Thus if 6 is infinitesimal, #, sin d, and tan 6 are equivalent; 
also an infinitesimal arc and its chord. 

If the limit of ft /a is h (h being any constant, not zero), 
the limit of ft /ha is 1, hence ft is equivalent to ha. Thus 
1 — cos 6 is equivalent to \6 2 , tan 6 — sin 6 to |# 3 . 

16. If the difference of two infinitesimals of the same 
order is of a higher order, they are equivalent. 

For, if p=a+i, ft/a = l+i/a, .'. £{ft/a) = l if £(i/a) = 0. 

Conversely, the difference of two equivalent infinitesimals 
is of a higher order. 

For, if ft/a = l+i, ft=a+ia, and ia is of a higher order 
than a. 

The letter / will be used as a symbol for higher infini- 
tesimals. Thus if 6 is infinitesmal, sin 6 = 6+1, tan 6=6 +I\) 
also, since 1 — cos 6 is of the second order, cos 6=1+I 2 - 

17. The limit of the quotient of two infinitesimals is not 
changed when either is replaced by an equivalent infini- 
tesimal. 

a ft' ' a' ' a' 

.'• £~=£^r if ^|=1 and £-,= !. 
a a ft a 

Hence if a and ft consist of infinitesimals of different orders, 
the limit of ft/a depends only on the infinitesimals of the 
lowest order in each. 

* Not equal, but equivalent in the sense of being interchangeable 
in the determination of the limit of a quotient or of a sum (§§ 17, 91). 



For 



15-17.] 



LIMITS. INFINITESIMALS. 



Examples. 

1. Let A B be a, circular arc of radius a subtending an infinitesi- 
mal angle 6 at the centre, BC perpendicular to OA, AD and 
BE tangents. Let be regarded as of the first order. Then 

(1) The arc AB=ad, and is therefore of the first order. 



. v ^chord AB _ J2a sin \Q 



e 



o 



=£ 2 -^ (§§15, 17) -a. 







Fig. 4. 



B^,D 




c A 



The chord is therefore of the first order and equivalent to ad, 
the arc. 

,«v r CA /• <*(! -cos 0) i 



: 



: 



Hence CA is of the second order and equivalent to %ad 2 . 



AD-CB q(tan 0-sin 0) 



— & . 2« 



(4) £ ^ 03 

Hence AD—CB is of the third order and equivalent to Ja0 3 . 

,*. r £D-CA r (CA/cos0)-CA r a(l-cosd) 2 
( 5 ) £ 71 — =£" 71 =£ 



6- 



=£ 



add 2 )' 



6- 



= \a. 



4 cos 



Hence BD —CA is of the fourth order and equivalent to \a0*. 

A T\ fl D 

2. Show that the limit of ^r— — — =2. 

3. Show that (AE + EB) —chord AB is equivalent to |a0 3 , and 
hence that the difference of arc and chord is an infinitesimal 
of at least the third order. 



4. Find £ 



X-2 
x 3 — x 2 — 2x 



whena;=l and x = 2. Arts. (1) J, (2) £. 



10 INFINITESIMAL CALCULUS. [Ch. I. 

Show that there is no limit when x = or —1. 
j. cosecz— cot x £ 1— cos 3 0_ 3 

u sinx sin 2 



(n -l)(w- 3) 

n(n— 2) 



7. £ n =oo ; 7T\ "~1« 8. £j;«— 00 (2-|-3 a; )— 2. 



v'l+a; —1 
9 - £c=o — = i- [Rationalize the numerator.] 

r sin a:— sin a 

10. £3. . n =cos a. 

x=a x-a 



CHAPTER II. 
FUNCTIONS. DERIVATIVES. DIFFERENTIALS. 

18. Function. When a variable quantity depends for its 
values upon those of another variable quantity, the first is 
said to be a function of the second, and the second is called 
the variable or argument of the first; e.g., x 2 — 2:r + l, x x , 
log (a+x), sin ax, are functions of the variable x. 

The expression f(x) is used as a symbol for a function of 
x, f(a) being the value of the function when x = a; e.g., 
if f(x) = l-x 2 , /(0) = 1, /(1) = 0, /(2)= -3, /(a)=l-a». 

For a similar purpose F(x), f(x), etc., may be used, and 
f(x, y), F(x, y), etc., for functions of x and y. 

The variable of a function may itself be a function of 
another variable, or it may be an independent variable — one 
to which arbitrary values may be assigned. 

19. Implicit functions. In any equation containing two 
variables x and y, e.g., y 2 =4ax, log (x + y) = 2, either of the 
variables is virtually or implicitly a function of the other, 
or an implicit function of the other, since the value of either 
is determined when that of the other is assigned. If we 
solve for y in terms of x, y becomes explicitly a function of 
x, or an explicit function of x. 

20. Graphs. The curve whose equation is y = f(x) is the 
graph or geometrical representation of the function /(#). 
The ordinate corresponding to any abscissa x is the value 
of the function when the value of the variable is x. 

When for a value of the variable there is only one corre- 
sponding value of the function, the function is said to be 

11 



12 



INFINITESIMAL CALCULUS. 



[Ch. II 



single- valued. Thus e x , Fig. 5, and log x, Fig. 6, are single- 
valued functions. The function sin -1 x, Fig. 7, is multiple- 
valued. It will in general be 
assumed that a function is 
single- valued ; when such is 
not the case the function may 
be treated as single-valued by 
considering a limited range 
of its values. Thus sin -1 x is 




single-valued 



if 



n 



values > -j: 



and < 



it 



— are excluded. The 



ordinate of the curve y 2 = 4x 

is a double- valued function of 

x, but may be represented by 

two single-valued functions 

2\/^and -2\/x. 

Fig. 6. Fig. 7. 21. Continuity. In general 

a gradual change in the value 
of a variable produces a gradual change in the value of the 
function, but it is possible that a slight change in the variable 
may produce an abrupt finite or infinite change in the func- 
tion. In more precise language, f(x) is continuous for the 
value a of the variable when, as /i = (h being a small change 

in x), 

£f(a + h) = £f(a-h) = f(a), 

and discontinuous if this relation is not true. 

LetOA = a,BA = AC = h. In Fig. 8, AP = /(a), CR = f(a + h), 
BQ = f(a-h). Also as ft=0, £CR=AP, and £BQ = AP, 
hence the ordinate is continuous at A. But in Fig. 9, 
£CR = AP, £BQ = AP'; these are not equal, hence y is dis- 
continuous at A. In Fig. 10, BQ becomes infinite when 
h = 0, and y is discontinuous. 

When the function changes abruptly from one finite value 
to another finite value it is said to have finite discontinuity, 



21-24.] 



CONTINUITY. DERIVATIVE. 



13 



when the function becomes infinite it is said to have infinite 
discontinuity. 






Ex. 1. /Or) 



has finite discontinuity at x =0. For, when 



2*+l 
h = 0, £}(h)=0 and £f(—h) = \. Thus when x increases through 
the value the function drops suddenly in value from slightly 
less than 1 to slightly more than 0, without passing through the 
intermediate values. It cannot be said to have any value when 
z=0. 

2. The following have infinite discontinuity for x = l: 

(x+l)/(x-l), (x-l)~i, 3^~^~\ tani^. 

3. Examine the function 2 X for x=0. 

22. Interval. Values of the variable between two assigned 
values a and b are said to lie in the interval from a to b. 
The interval may be conveniently indicated by [a, b]. Re- 
versal of a bracket indicates the exclusion of the adjacent 
end value; e.g., [a, b[ indicates values from a to b, including 
a but excluding b. Thus (1 — x 2 ) 1 * is real for the interval 
[— 1, 1], (1 — z 2 ) - * is continuous for the interval ] — 1, 1[. 

23. Increment. Any change in the value of a quantity 
is called an increment or difference of that quantity. An 
increment is positive or negative according as the quantity 
is increased or decreased. The symbols Ax, dx, are used 
for increments or differences of x. 

24. Derivative. Let there be a variable and a function of 
that variable. A particular value of the variable being x 



14 INFINITESIMAL CALCULUS. [Ch. II 

let the corresponding value of the function be y. Let x 
receive an increment Ax, and let Ay be the corresponding 
increment of y. The limit of Ay/ Ax when Ax = is called 
the derivative of y. Thus the derivative is the limit of the 
quotient of the increment of the value of the function by 
the increment of the value of the variable when the latter in- 
crement is infinitesimal. The primary object of the Infini- 
tesimal Calculus is to determine this limit for various kinds 
of functions. 

Ex. 1. liy=x 2 , Jy = (x+Jx) 2 -x 2 =2xJx+(Jx) 2 . 
.*. Jy/Jx=2x + Jx. .'. £{Ay/Ax)=2x when £Jx=0. 
Similarly, if y=ax 2 , £(Jy/Jx) =a . 2x, a being any constant. 

2. If y =ax 3 , show that £(Jy/Jx) =a. 3x 2 . 

3. If y=x, Ay = Ax, .'. Ay/Ax = \, or £(Ay/Jx)=l (§5). 
Similarly if y=ax, £{Ay/Ax) =a. 

4. y=4x 5 -3x 2 + 2x-l. 

The method of obtaining Ay /Ax shows that the result will be 
the same as if each term were treated separately and the results 
added, also that a constant term disappears in subtracting. 

/. £(Jy/Jx) =4: .3x 2 -3 .2x + 2 = 12x 2 -6x + 2. 

25. The general method illustrated in the above examples 
may be described as follows: Let y=f(x). The value of the 
function for the value x-\-Ax of the variable is }(x + Ax); 
hence Ay, the increment of function, is f(x + Ax) — f(x), and 

Ay __f(x + Ax) — f(x) 
Ax Ax 

This expression is simplified and its limit taken when Ax = 0. 
This limit is the derivative, and is, for various values of x, 
a new function of x. It is called the derived function, or 
derivative function, or simply the derivative, of the given 
function. Let it be written f'(x). Thus if f(x) = x 2 , f'(x) = 2x; 
if./(x) = ax 3 , f'(x) = 3ax 2 ; if f(x) = x 2 + 2ax, f'(x) = 2(x + a). 

It should be noticed that there cannot be a limit (a deriv- 
ative) unless Ay = as well as Ax; i.e., unless £f{x J rAx) = j(x) 
when Ax = 0, or (§ 21) unless ](x) is continuous for the value 
of x in question. 



25-27.] 



DIFFERENTIAL. 



15 





Y 












a 








p^£ 


Ay 






Ax 




^$ 




y 




y 



o a? Aa? 
Fig. 11. 



26. Geometrical illustration. Let y = f(x) be the equation * 
of a curve of which P(x, y) is a 
point. Then Ay/ Ax is the slope or 
gradient of the secant PQ. When 
Ax^O, Q approaches P, and the limit 
of pc sit ion of the secant is (by defi- 
nition) the tangent at P. Hence 
£(Ay/Ax) or f(x) is .tan 0, the slope 
of the tangent at (x, y). Thus for 
the curve y = x 2 , tan<£ = 2:z; for y=x 3 , tan<^> = 3x 2 . 

27. Differential. Def. The differential of the variable of 
a function is any increment of that variable; the differential 
of the function is the derivative of the function multiplied 
by the differential of the variable. 

The letter d is used as an abbreviation for "the differential 
of." If then y or f(x) is a function of x, the definition states 
that dx is any increment of x, and that dy or df(x) is fix) dx.~\ 



Hence 



g=f(s) = 4>en^0. 



Thus dx and dy are defined in such a way that dy/dx is equal to 

the limit of Ay/ Ax, or the differential quotient is the limit of 

the difference or increment-quo- 
tient. 

Geometrically, dy/dx is the 
slope X of the tangent at (x, y), 
Fig. 12, and dy is the increment 
of the ordinate of the tangent 
corresponding to the increment 
dx of the abscissa. It should be 

noticed that, although dx may have any value, the value of 

dy/dx is independent of dx. 

* The angle between the axes is assumed to be a right angle in all 
cases unless the contrary is mentioned. 

t From its position as a multiplier of dx the derivative f\x) is also 
called t e differential coefficient of fix). 

% If the angle between the axes is co, dy/dx = sin <£/sin (<o — </>). 




CHAPTER III. 
DIFFERENTIAL OF A POWER, A PRODUCT, AND A QUOTIENT. 

28. The operation of obtaining derivatives or differentials 
is called differentiation. 

We now consider a few general formulae which will assist 
in differentiating, first showing that the differential of the 
algebraical sum of any finite number of terms is equal to 
the algebraical sum of the differentials of the terms; also 
that a constant factor in a term appears as a factor in the 
differential of that term, and that a constant term dis- 
appears in differentiating, or has for its differential. 

Let y=au + v — w + c, where u, v, w are continuous func- 
tions of x, and a and c are constants. Let the increment 
Ax in x cause increments Au, Av, Aw in u, v, w, Ay being 
the resultant increment of y. Then 

Ay = [a(u + Au) + (v + Av) — (w + Aw) + c] — (au +v — w+c) 
= a Au+Av — Aw. 

Ay _ Au Av Aw 
Ax Ax Ax Ax' 

Hence, taking the limits when Ax±=0, 

dv du dv dw , 77, 

-r £ = a- r -+- —, or dy=a du-\-dv — dw. 

dx dx dx dx 

.'. d(au + v — w + c) = a du+dv — dw. 

29. Differential of a power. Let v be a function of x, to 
find d(v n ), n being any constant. Let y = v n . Then 

Av\ n 



/ dv\ n 
Ay=(v+Av) n -v n = v n (l-\ — ) -1 



16 



28, 29.] POWER. PRODUCT. QUOTIENT. 17 

Taking Av\<\v, expanding by the binomial theorem, and 
dividing by Ax, 



Ay - Av 

^ nv n- 



I ' n—\ Av \ 



Ax Ax 

Taking the limits when Ax^O, and .'. Av also = 0, 

dy , dv 

dx dx J 

or d (y n ) = nv n ~ l dv. (A) 

This result is true for all values of n. The case in which 
n = \ deserves special mention; (A) then becomes 

d(V?)« * (B) 

2Vv 

When n= — 1, (A) becomes 

Ex, 1. d(x 5 )=5x*dx. 

2, d(3x 5 + 2)=3d(x 5 )=l5x*dx. 

3 d(3x^_-2x 2 + 6)=3 Ax 3 dx-2.2xdx=4x (3x 2 -l)dx. 

4. dV x 2 =d(x*) =%x~$ dx. 

5. d(-) =d(x~ 2 )=-2x- z dx= -. 

\x 2 / X 3 

6* d(a 2 + x 2 y=3(a 2 + x 2 ) 2 d(a 2 + x 2 ) 

=S(a 2 + x 2 ) 2 2xdx=6x(a 2 + x 2 ) 2 dx. 
In this example v=a 2 + x 2 , and n=3, a being constant. 



„ j/a—xX 1 7, x dx 

7. a I = -d(a—x)= -. 

\a—b/ a—b a — b 



8 d . =d(ax + bx 2 )~* 

v(ax + bx 2 ) 3 

= -%(ax + bx y-r* d(ax + bx 2 )= -Uax + bx 2 )-Ha + 2bx) dx. 



n W"l 2 d{a 2 -x 2 ) /T>N -2xdx x dx 

9. dv a *-x 2 = — - \ by (B), = — = 

2V a 2 -x 2 2Va 2 -x 2 v 'a 2 - 

( 1 \ d(a 2 -x 2 ) 2sds 



18 INFINITESIMAL CALCULUS. [Cn. III. 

30. Differential of a product. Let y=uv, where u and v 
are functions of x. Then 

Ay = (u+Au)(v + Av)—uv = v Au+u Av+Au Av. 

Ay Au Av . Av 
.'. -r = v—+u-7-+Au—. 
Ax Ax Ax Ax 

The limit of the last term is 0. 

dy du dv 
• • 7 v^ 1 t^ ^ . 
ax ax ax 

or d(uv) = v du + u dv. (C) 

Similarly, d(uvw) = vw du+wu dv +uv dw. (C^) 

Ex. d(4x + 3)(x 2 -l) = (x 2 -l)d(4:X + 3) + (4x + 3)d(x 2 -l) 

= (x 2 -l)4:dx + (4:x + 3)2xdx=2(6x 2 + 3x-2)dx. 

31. Differential of a quotient or fraction. Let the frac- 
tion be u/v, u and v both being variable. Then 

d ^) =d \ U v) = v du+U \ : ^)' by (C) and (Bl) * 

TT Ju\ vdu — udv /T ^ X 

Hence d[ — )= 5 . (D) 

_ 7 /V-l\ (x 2 + l)d(^ 2 -l)-(x 2 -l)d(o: 2 + l) 4zdr 
.Lx. d 



x 2 +V (x 2 + l) 2 (z 2 + l) 2 ' 

Examples. 

1. d(a 2 — x 2 ) 3 = — 6x(a 2 — x 2 ) 2 dx. 

2. d^\Vx 2 =xdx/^\+x 2 . 

3. If j(x)=ax 2 + 2bx + c, f'(x) =2(ax + b)* 

4. y = Vx' d —a 3 , dy/dx=3x 2 /2\ // x 3 —a 3 . 

5. d [ax(x 2 - l)(x + 1)] = a(x + l)(4o: 2 -3 - l)dc. 

/^-1\ 8a: 3 da; 

7. d(l+x)\ / l-a; = (l-3a;)da;/2\ / l-a;. 

*A*)-^/(*)/*i|27. 



30,31.] POWER. PRODUCT. QUOTIENT. 19 

8. y = 3(a + bx 2 )*, dy^lObx(a + bx 2 )sdx. 

9. y = vV + {b -x) 2 , dy=(x- b)dx/V a 2 + (b -x)\ 

0. y={a 3 -x 3 )~\ dy/dx = 3x 2 /(a 3 -x 3 ) 2 . 

1. i/ = o; 3 /(«+^) 2 , dy/dx= (3a+x)x 2 /(a+x) 3 . 

2. y=\/a 2 —x 2 ,dy=—xdx/^a 2 —x 2 . 



3. y=^2ax—x 2 , dy=(a—x)dx/v / 2ax—x 2 . 

4. y = x*/(a 2 -x 2 ), dy/dx= (3a 2 -x 2 )x 2 /(a 2 -x 2 )\ 

5. d[z n /(l +x) ri ] = nx^- 1 dz/(l +x) ri + l . 

6. d(a 2 -z 2 )- 1 = 2xda;/(a 2 -a; 2 ) 2 . 

7. i/=a; 2 /v / l+x 4 , dy/dx = 2x/(l+x 4 )*. 

8. 2/ = (x-a)/v / x, dy/dx = (a + x)/2Vx*. 

9. y=ax/\ // 2ax—x 2 , dy/dx=a 2 x/(2ax—x 2 )*. 

on \ aJrX j adx 



a ~* (a-a;)\/a 2 -x 2 



21. 2/=2x/Va 2 + z 2 , d?/=2a 2 ^/(a 2 + x 2 )i. 

22. .y=z(a 3 + z 2 )>/a 2 -x 2 , cfy/dx = (a 4 + a 2 x 2 -±x 4 )/^a 2 -x 2 . 

23. f(x)=V x + Vl+x 2 , J'(x)=Vx + Vl+x 2 /2Vl+x 2 . 

x 2 y 7, 

24. — + --=1. Differentiating each term, 

2x dx 2y dy , dy b 2 x 

a 2 b 2 ' ' ' dx a 2 y 

25. y 2 =£ax, dy/dx = (a/x)*=2a/y. 

26. x 2 y + b 2 x-a 2 y=0 y dy/dx = (b 2 + 2xy)/(a 2 -x 2 ). 

27. x 3 + y 3 =3axy, dy/dx= — (x 2 —ay)/(y 2 —ax). 

28. x 2 y—xy 2 =a 3 , dy/dx = (y 2 —2xy)/(x 2 —2xy). 

29. If y-i show that ^_ + — ^==0. 

* Vl+z 4 Vl + ^ 



CHAPTER IV. 



TANGENTS AND NORMALS. 



32. Let P and Q be two points near one another on a 
curve of which the equation is given. Let the coordinates 
of P be (x, y), then x = OA, y = AP. When x has the in- 



Y 




F 


/4i 


/ 




c 




/<& 










Fig. 13. Fig. 14. 

crement Ax or AB, the new value of y is BQ, hence CQ is Ay. 
Let the tangent at P make an angle <f> with the x-axis. Then 
as in § 26, when Q approaches P as a limit of position, Ax = 0, 
and 

tan (j) = £ tan CPQ = £(Ay/Ax) = dy/dx. 

Let the length of the curve measured from some point up 
to P be s, and let the length of the arc PQ be As, and the 
length of the chord PQ be q. Then 

cos <f> = £ cos CPQ = £(Ax/q) = £(Ax/ As) (§17) =dx/ds. 

Similarly, sin cf) = dy/ds. 

Thus, cos^ = g(l), sin<£ = |j (2), tan«£=g (3). 

20 



32, 33.] 



TANGENTS AND NORMALS. 



21 



or 



Squaring (1) and (2) and adding, 
* ds 2 = dx 2 + dy 2 . 





o N A 



Fig. 15. 



Fig. 16. 



These relations show that if dx is PD (Figs. 15, 16), dy 

is DE, and ds is PE. 

dx ds 

33. The subtangent TA = y—, the tangent^ TP=y—, 

the subnormal AN = y -p-, the normal NP = y -r-. 
The intercepts of the tangent on the axes are 

OT = x-y^, OS= -OT tan <f> = y-x& 



y - y *= (fX {x ~ xi) 



Also, 
is the equation of the tangent, and 

* Powers of a differential dx are written dx 2 , dx 3 , etc. They must 
be distinguished from d(x 2 ), d(x 3 ), etc., which are differentials of 
powers of x. 

t The line-segments known as the tangent and normal are the 
portions of the tangent and normal which join the x-axis to the point 
of contact. 



22 



INFINITESIMAL CALCULUS. 



[Ch. IV. 



the equation of the normal, at a point whose coordinates 
are (x\, 2/1), the parentheses ( ) x indicating the particular 
value which the enclosed quantity has when x\ and y\ are 
substituted for x and y. 

It will be convenient to take dx as +, i.e., measured in 
the + direction of the a>axis, and to suppose (p to be a 
positive or a negative acute angle; hence cos and ds are 
always + , and sin <fi and tan <j> have the same sign as dy. 
It should be noticed that the curve rises or falls (y increases 
or decreases) according as dy is + or — . 

Examples. 

1. The curve a 2 y=x(x 2 —a 2 ), Fig. 17. 
Differentiating we have tan <i>=dy/dx = (3x 2 —a 2 )/a 2 . 
At the origin x=0, .'. tan <f>= — 1 and .'. (f>= —45°. 
At A or 5, x= ±a, .'. tan <£ =2 and <j> =63° 26'. 
When x= ±a/Vs, tan <£=0, /. <£=0. 








Y 











-— ^ 






y 

/ 










/ 
/ 


/S 




\ 

\ 




/ 

/ y 






V \ 




1 / 










l"^s 







/ 1 


X 


\ 






/ 




\ 






/ 




\ 






/ 




X 











Fig. 17. 



Fig. 13. 



The equation of the tangent at any point (x ly y x ) is 



y-y* 



2. The common parabola y 2 =4ax. 

Differentiating each term, 2y dy =4:0, dx, .'. dy/dx=2a/y. 

.'. y —y l = {2a/y l ){x—x l ) is the equation of the tangent at 
(x lf y x ), and reduces to y 1 y=2a(x + x 1 ). The subnormal =y dy /dx 
= 2a, a constant. 

3. The equation x% + y*=a$ represents an astroid, or four-cusped 
hypocycloid (Fig. 18), i.e., the locus of a point in the circum- 



33.] TANGENTS AND NORMALS. 23 

ference of a circle which rolls inside the circumference of a fixed 
circle, the diameter of the latter being four times that of the former. 
Differentiating the equation, we get %x~$ dx + %y~% dy=0, whence 
dx/dy = — (x/y)$. The intercepts of the tangent on the axes 
will be found to be cfi x% and a$ y$. Squaring, adding, and taking 
the square root we find that the part of the tangent intercepted 
between the axes is of constant length, viz., a. Hence, if a straight 
line of length a slide with its extremities on two given lines at 
right angles to one another, it will constantly touch this curve. 

4. To find tan <$> at any point of the curve x 2 y—xy 2 =2. 
Differentiating each term by (C), 

x 2 dy + 2xy dx—2xy dy—y 2 dx=0, 

.*. dy/dx = (y 2 —2xy)/(x 2 —2xy). 

5. Find the equations of the tangents at the points (—1, 1), 
(2, 1) on this curve. Arts. x—y + 2=0, x=2. 

6. Of the rectangular hyperbola xy=k 2 show that 

(1) the equation of the tangent at (x 1} y x ) is x/x 1 -{-y/y l =2 f 

(2) the equation of the normal at (k, k) is y=x } 

(3) the subtangent always = —the abscissa, 

(4) the tangent makes with the axes a triangle of constant 
area, viz., 2k 2 . 

7. Show that the tangent to the curve (x + a) 2 y=a 2 x is parallel 
to the axis of x when x=a f perpendicular to it when x= — a, and 
that the tangent at the origin bisects the angle between the axes. 

8. Find the equations of the tangent and normal at the point 
(a, a) on the curve ay 2 =x 3 . Arts. 3x—2y=a, 2x + 3y=5a. 

Also show that the subtangent = fa, the subnormal = fa, the 
tangent =^aVl 3, the normal =JaVl3. 

9. On the curve x 2 y + b 2 x=a 2 y, show that tan 4>=b 2 /a 2 when 
x=0, 20b 2 /9a 2 when x=^a y and 5b 2 /9a 2 when x=2a. 

10. Show that the curves y(4: + x 2 )=8, 4y=x 2 , intersect at an 
angle tan -1 3. 

11. Find the equations of the tangents of the following curves 
at the given points : 

(1) xy = l+x*eLt (1,2). Arts. x-y + l=0. 

(2) x 2 + y 2 =x s at (2, 2). 2x-y=2. 

(3) xn + yn=xfi+ l &t (2, 2). (n + 2)x-ny =4. 



24 



INFINITESIMAL CALCULUS. 



[Ch. IV. 



(4) a 2 y=x* at (x u y x ). 

(5) y 2 =3x+l at {x l} y x ). 

a 2 o 2 



Sa^lc — a 2 !/=2:r, 8 . 
3^-22/^ + 3^4-2=0. 



a 2 6* 



the 



12. 4:xy=4: + x 3 ; show that at (2, f) the subnormal = f, 
subtangent =2, the normal = V, the tangent = f. 

13. y 2 =3x + l; show that when y=—£ the subnormal = f , 

the subtangent =-3-, the normal = — JV73, the tangent =1^73. 

14. x$ + y$=a$, Fig. 18; show that ds = (a/x)$dx. 

15. Find an expression for the length of the perpendicular 
from the origin on the tangent at any point (x, y) of any curve. 

Ans. (x dy—y dx)/ds. 

16. In the case of the parabola y 2 =4:ax, show that the length 

of this perpendicular = x Va/ (a + x). 



CHAPTER V. 
DIFFERENTIALS OF EXPONENTIALS AND LOGARITHMS. 

34. Differentials of the exponentials a* and e v . Let a 

be any constant, v any function of x. Then 

A(a v ) = a v + jv -a v = a v (a* v -l) 

= a*[A Av + J A 2 (Av) 2 + ...], A = loge a, 

by the exponential theorem, the series being convergent for 
all values of Av. 

Ax Ax 

and taking the limits, 

d(a v ) _ a v dv 
dx dx 1 

or d (a v ) = Aa v dv. (E) 

When a = e = 2.71828 . . . , A is 1. Hence 

d«) = ^cfa. (F) 

Ex. 1. d(e 3x ) = e 3x d(3x) = 3e 3x dx. 

2. d{e~ x ) = e~ x d( —x) = —e~ x dx. 

3. d(2~ x ) = (loge 2)2-* d( -*) = - (loge 2)2"* dx. 

35. Differentials of logarithms. First suppose the loga- 
rithms to be Napierian (or hyperbolic or natural) logarithms, 
the base being 6 = 271828 . . . Let y = \og e v, 

25 



26 INFINITESIMAL CALCULUS. [Ch. V. 

then v = e v , .'. dv=e v dy=v dy, 

dv 
.*. dy, or d(\og e v) = — . (G) 

Secondly, let the base be any number a. Then V log a v = 
M loge v, where M= 1/loge a, or = loga e, 

.-. d(\og a v) = M— . (Gi) 

V 

M is the modulus of the system of logarithms with base a. 
Unless the contrary is indicated, the logarithms are always 
assumed to be Napierian. 

Ex. 1. dlog (ax 3 )=d(ax*)/ax z =3 ax 2 dx/ax 3 =S dx/x. 
2. d(x log x) =log x dx-Vx d(log x) = (log x + l)dx. 

36. To differentiate u v , where both u and v are variable 
quantities. Let y = u v , then log y= (log u)v; hence, differen- 
tiating, 

— = v \- (log u)dv, .'. dy = y\v h (log u) dv \ 

y it i— tc -J 

.*. d(u v ) = v u v ~ l du+ (log u)u v dv, (G 2 ) 

i.e., the differential is obtained by supposing u and v in 
turn to vary while the other remains constant, and adding 
the results. 

37. When an expression is made up of factors it is often 
simpler to take logarithms before differentiating. 

Ex. 2/ = (z + l)*(x + 3)V(z + 4) 4 , 

logy=ilog (x + l)+f log (:r + 3)-41og (x+4), 
dy 1 dx 9 dx t dx 



y 2 x + \ 2x + 3 x + 4' 
whence dy. 



36,37.] EXPONENTIALS AND LOGARITHMS. 27 



Examples. 



1. y=- log— — = — [log (a; -a) -log (x + a)], dy = 



2a x + a 2a x 2 —a 



\—i 



2. y=\og (x/Vl + x 2 ) =log x— \ log (1+x 2 ), dy=dx/(x + x 3 )> 

3. }(x) = Vx- \og(l + Vx), f , (x)= j(l + y/x)- 

4. dlog (z + V^ia^-dx/V^ia 2 . 

5. y=\og[e x /(l+e x )], dy=dx/(l+e x ). 

6. y=e xn , dy=ne xn x n ~ 1 dx. 



7. dlog (Vx-aH- V / x-6)=idx/v / (a:-a)(a;-6)» 

8. y= a loex = (e A ) log * = (e log *) A =x A , <fy =a x^~ l dx. 

9. d(x :c ) = (l+log x)x x dx. 

10. i/ = l+x^, dx = (2-y)dy/e v . 

11. y=log (i//:r), (l-y)xdy=ydx. 

12. Differentiate y=uv> y=uvw, and y=u/v, after taking 
logarithms, and compare the results with formulae (C), (CJ, and (D). 

13. Differentiate y=v n after taking logarithms, thus showing 
that d(v n ) =nv n ~ l dv. 

14. Show that the subtangent of the exponential curve y=a x 
is constant and =log a e. 

15. Find the subnormal of the curve y 2 =a 2 log x. 

Arts. a 2 /2x. 



Ad 


-fc 


Ad 


d sin v 
dd 


= cos V 


dv 
dd' 


d sin v = 


= cos V 


dv. 


d cos v = 


■■ — sin v dv. 



CHAPTER VI. 

DIFFERENTIALS OF DIRECT CIRCULAR (TRIGONOMET- 
RICAL) FUNCTIONS. 

38. To differentiate sin v. Suppose v to be a function of 
6. Then A sin v = sin (v + Av) — sin v = 2 cos (v + \Av) sin £ Jv,* 

. r i sin v r 2 cos v . \Av /ff v _ 
•-£—nr- = £ 7^ > (§ 17 ) 



or 



(1) 
Similarly, d cos v = — sin v rfy. (2) 

39. The differentials of the remaining functions may be 
found by first expressing them in terms of sine and cosine 
The results with (1) and (2) above are: t 

d sin v = cos v dv, (H) 

d cos v = — sin v dv, (I) 

d tan v = sec 2 v dv, (J) 

d cot v = — cosec 2 ?; <#y, (K) 

d sec v = sec v tan 7; cfa, (L) 

d cosec v = — cosec v cot v dv. (M) 

* Since sin A— sin # = 2 cos i(A + i?) sin h(A— B). 

t To these results may be added: 

d vers v = d(l —cos r)=sin v dv, 
d covers v = d(l— sin v)= —cos v dv. 

28 



B3,39.] DIRECT CIRCULAR FUNCTIONS. 29 

Examples. 

1. d sin nO = c(^s nO d(nO) =n cos nO dO. 

2. d sin (tan 0) = cos (tan 0) d(tan 0) =cos (tan 0) sec 2 dO. 

3. y=tan0-0, dy = tsa\ 2 ddd. 

4. d(£0-isin20)=sin*0d0. 

5. d(i0 + isin20)=cos 2 0d0. 

6. d(sec + tan 0) = (l+sin 0) d0/cos 2 0. 

7. y=J tan 3 £ — tanx+xj dy=t2Ln 4 xdx. 

8. /(a;) = sin a: —J sin 3 a;, /'(a;) = cos 3 x. 

9. d(sin 2 .r cos 2 x) = £ sin 4x cfo. 

10. ?/ =log tan J0, dy= cosec dO. 

11. 2/= log tan (i?r + £0), dy=secdd0. 

12. d(sec + log tan £0) = sec 2 cosec d0. 

„_ ./ sin x \ (cos 3 x—sm^x)dx 

13. d[~ I =-—. — . 

U + tanov (sin a: + cos xy 

14. /(x) =sin (log x), f'(x) =x~ l cos (log x). 

15. de* cos x =e x (cos x —sin x) dx. 

16. d log sin = cot d0. 

17. d log cos = —tan d0. 

18. d log tan =sec cosec d0. 

19. d log sec 0=tan dO. 

20. d log (sec + tan 0) =sec d0. 

21. y =log Vsin x + log Vcos x, dy/dx = cot 2x. 

22. y=2/(l+tanix), dy/dx = -1/(1+ sin z). 

23. rilogV(l-cos 0)/(l + cos 0) = cosec Odd. 

24. d sin nd sin n = n sin n_1 sin (n + 1) d0. 

25. By differentiating sin 20=2 sin cos 0, show that 

cos 20=cos 2 0-sin 2 0. 

26. The cycloid. This is the curve traced by a point in the 
circumference of a circle which rolls along a straight line, Fig. 19. 

Let 0=the angle through which the circle (of radius a) rolls 
while the tracing-point moves from to P. 

Then 

x = OM = OB-MB=2lyc PB-PD = ad-asmd, 
y = MP = BC-DC = a -a cos 0. 

From these two results may be eliminated; but as the result- 



30 



INFINITESIMAL CALCULUS. 



[Ch. VI, 



ing equation is not algebraical we shall suppose the locus deter 
mined by the simultaneous equations 

x=a(0 — sin 0), y =a(l — cos 6). 




o M 



Fig. 19. 



For a single arch of the curve varies from to 2n\ for greater 
or smaller values of the curve is repeated indefinitely in both 
directions. 

Produce BC to meet the circle in E 1 then PE is the tangent 
and PB the normal at P. For 

dx=a(l -cos 6) dd =BD dd, dy=a sin 6dd=DP dd, 
/.if the tangent makes an angle (j> with the axis of x, 
tan <f>=dy/dx=DP:BD=tfmDBP = ttmDPE } 

BPE m an angle in a semicircle, being a right angle. Therefore PE 
is the tangent. Hence at each instant the circle may be supposed 




to be turning about its lowest point as an instantaneous centre 
of rotation." 

Since CE = CP,CEP=W J ,\ the normal PB = 2a sin £0. 



30.] DIRECT CIRCULAR FUNCTIONS. 31 

27. If the axes of the cycloid be taken as in Fig. 20, its equa- 
tions are 

x=a(l — cos 0), y =a(#-fsin 0), 

being the angle through which the circle has rolled from R. 

The locus of Q (of D in Fig. 19) is called the " companion to 
the cycloid." Its equations are 

x=a(l — cos 0), y=aO. 

Show that in this curve tan <£=cosec 0, and hence that <£ is least 
when x=a. 

28. At any point of the cycloid, show that 



ax y y y 



dy \2a „ V2ax— x 2 • 
29. In the cycloid, Fig. 20, show that 



ds/dx = \/(2a)/x. 



CHAPTER VII. 

DIFFERENTIALS OF INVERSE CIRCULAR (TRIGONOMET- 
RICAL) FUNCTIONS. 

v 
40. To differentiate sin -1 — the radian measure of the 

a 

angle whose sine is v/a, a being a constant. 

v 
Let y=sin _1 — , then v = a sin y. 

I v 2 / 

c°. dv = a cos y dy=a^l — ^ dy=V a 2 — v 2 dy. .'. dy or 

v dv ^ dv 

d sin" 1 - = , (N), or dsin~ 1 v = . , if a=l. 

a vV-v 2 Vl-v 2 

Similarly, 

, _J dv , dt> .. 

a cos x -= (Ni), or d cos 1 v= . . if a=l: 

a Va 2 -v 2 Vl-v 2 

dtan -1 - = -^- — ^ (P), or d tan -1 i> = T — —5, if a=l; 
a cr + ir 1+v 2 

, . J a dv - dv 

a cot l ~ = — 5-7 — 9 (Pi), or d cot 1 v= — — — 5, if a=l; 

a a 2 + v 2 1+v 2 

v adv ^ , dv 

dsec x -= — , (Q), or d sec x v = — ,. , if a=l; 

a v V ^2 _ a 2 ^\/ ^2 _ I 

7 , v a dv ,~ N 7 1 dv 

d cosec -1 - = (Qi)> or d cosec _1 v= . , 

a v \/ v 2_ a 2 -yVv 2 — 1 

if a=l. 

* This formula should be preceded by a minus sign if cos?/ is — , 
i.e., if the angle is a second or third quadrant angle (see Fig. 7). The 
f rmulse as given may be supposed to apply only to first-quadrant angles 

32 



40.] INVERSE CIRCULAR FUNCTIONS. 33 



1. dsin- 1 (2x 2 ) = 



Examples. 
d(2x 2 ) 4x dx 



Vl-(2x 2 ) 2 Vl-4x 4 

dtan-^ dx 

2. d(log tan _1 x) = 



tan _1 x (l+x 2 ) tan _1 x" 

6x da; ,1 j , a a ^ 

3. a sin -1 3x 2 = — , 4. d cos -1 — = 



V1-9X 4 a; x vV-a : 

_ , . x— a a dx _ , , a— a; dx 

5. asm -1 = — . 6. a cos -1 



^ xV / 2ax— a 2 a V2ax— x 2 

m , _, x 2 2x dx dx 

7. a sin l — = — - . 8. d tan -1 ^ = 



a 2 vV _£4 e* + e~* 

9. d sec- 1 Vl+x 2 = dx/(l +x 2 ). 

10. d tan- 1 (Vl+x 2 - x ) = -\dx/{\ +x 2 ). 

11. dsin- 1 (x/\ / T+x 2 )==dx/(l+x 2 ). 

12. d sin- 1 [(1 -x 2 )/(l H-x 2 )] = -2dx/(l+x*). 

13. d tan- 1 [2z/(l -o; 2 )l = 2d:c/(l + 2 2 ). 

14. d sin-^sin a;=£Vl +cosec a; dx. 

15. d vers -1 — =d cos -1 (1 ) = — — 

a \ a/ \Z2ax-x 2 



U-a da; 

16. a sin -1 



&-a 2V(x-a)(6-x)' 

17. y=asm- l (x/a) + \ / a 2 —x 2 , dy=dx\/(a—x)/(a + x). 

18. i/ = \/x 2 — a 2 — a sec -1 (x/a), dy=dxVx 2 —a 2 /x. 

19. i/= a; tan _1 a; — log Vl+x 2 , dy=ta,n~ l x dx. 

on * i .1 * ^ (l+x)dx 

20 o y=tan- 1 x + log^=, ^-^jj+^y. 

x — a x dy 2ax 2 

21 2/=log v — — + tan~ 1 -, 



x + a a dx x 4 —a* 



CHAPTER VIII. 



DIFFERENTIALS OF HYPERBOLIC FUNCTIONS. 



41. Def. The quantities \{e x — e~ x ), \(e x + e~ x ) are called, 




Fig. 21. 



respectively, the hyperbolic sine (sinh x *) and hyperbolic 
cosine (cosh x) of x. The hyperbolic tangent cf x is defined 



* This may be read "sine h of 2." 



34 



41 J HYPERBOLIC FUNCTIONS. 35 

i 

to be sinh .r/cosh x, and the hyperbolic secant, cosecant, 

and cotangent to be the reciprocals of the cosine, sine, and 

tangent, respectively. 

The graphs of the functions are represented in Fig. 21. 
Observe that sinh x may have any valua, cosh z^ 1, 
tanhx> — 1 and <1, coth:r>l or < — 1, etc., sinh = 0, 
cosh0= 1, etc. 

The fundamental relations 

cosh 2 x — sinh 2 £ = 1 , sech 2 a; = 1 — tanh 2 z, 
cosech 2 a; == coth 2 £ — 1 , 

are easily verified. 

The differentials of the hyperbolic functions are similar to 
those of the circular functions. Only the most important 
are given here. (For the others see Appendix, Note C.) 

Differentiating sinh v = \(e v — e~ v ) y co^\iv = ^{e v + e~ v ) } we 
have 

d sinh v = cosh v dv, 

d cosh v = sinh v dv, 

whence may be deduced 

j • i i v dv 

a sinh x - = 



a Vv 2 +a 2 ' 

j i i v dv 

a cosh 1 - = 



<*> \ v 2 -a 2 
v a dv 
a a 2 — v A 



dtanh i-==— — v\<]a, 



7 ,i _i v adv . - 

dcoth 1 -^-^ -, v\>\a. 

a a 2 — v 2 l ' 

Examples. 

1. y =log cosh x, dy/dx =ta,nh x 

-* x < u x x j dx Ix + a 

2. y = sec J — + cosh~ 1 — , dy=-.\ . 

* a a xyx-a 



36 INFINITESIMAL CALCULUS. [Ch. VIII. 



3. y=xVa 2 + z 2 + a 2 sinh- 1 (x/a), dy/dx=2Va 2 + x 2 . 

4. Show that sinh -1 — = log( ), 

a \ a / 



tanh~ 1 -=-log ( ). 

a 2 \a—xl 



[Let smb.' 1 x/a=z and e z =u. Then #/a=sinh z=%(u—ur l ). 
Solve for u in terms of x.] 

5. Gudermannian. If x = log tan (in + id), or log (sec + tan0), 
6 is called the gudermannian of x (gd x) and x is gd _1 0.* Prove 
that 

d gd x=sech x dx } 

d gd _1 x=sec x dx. 

[Differentiate, gd # =2 tan -1 ^ — in, and 

gd -1 £ =log (sec x + tsm x).] 

6. If z=log (sec + tan d), prove that 

cosh x =sec 0, sinh x =tan #, tanh a: = sin 0. 



* The inverse gudermannian gd —1 # is also written X{6), i.e., 
A(#)=l g tan (i^+ J<9)=log (sec 0+tan 6). 



CHAPTER IX. 



DIFFERENTIALS AS INFINITESIMALS. 

42. Let y be a function of x, dx an increment of x, and 
suppose y and its derivative to be continuous from x to 
x + dx. Let Ay, dy, be the increment and differential of y 
corresponding to dx. Let dx 
become smaller and smaller and 
« 0, then Ay and (in general) dy 
are also infinitesimals. Since 



Ay _dy 
^dx~dx' 



Ay 
dx 



dy 
dx 



+h 




where i is infinitesimal. Hence 

Ay = dy + I, (1) 

where / is an infinitesimal of an order higher than that of 
dx and dy. 

If dy^O, l/dy = 0, and / becomes a very small part of dy. 
Hence dy, when very small, is a close approximation to Ay. 
In reality (1) implies that dy is what remains of Ay when 
the higher infinitesimals are omitted; in other words, if 
higher infinitesimals are left out of account dy may be used 
as if it were the increment of y corresponding to the incre- 
ment dx of x. 

If 2/ = /Or), (1) may be written 



f(x + dx) - f(x) = f {x)dx + 1, 



(2) 



where l/dx = 0, and hence / is a very small part of dx when 
dx is very small. 

37 



38 



INFINITESIMAL CALCULUS. 



[Ch. IX 



43. Differentiation by the omission of the higher infini- 
tesimals is much used in the practical applications of the 
subject, and may be illustrated by the following examples. 
It must be remembered that dx is now regarded as infini- 
tesimal, and that the higher infinitesimals are not omitted 
because they are of trifling numerical value, but because 
they do not affect the final limit expressed by dy/dx. (See 
§ 17.) The result is in no sense an approximation. 

Ex. 1. If y=x n , Ay = (x + dx) n —x n 

= x n + nx n ~ 1 dx + . . . — x n =nx n ~ l dx + . . . , 

the terms indicated by . . . being higher infinitesimals. When 
these terms are omitted A changes into d. 

.'. d(x n ) =nx n ~ 1 dx. 

2. y=e x , Ay =e x + dx -e x = e x (e dx -1) =e x (l+dx + . . .-1). 

.'. dy=e x dx. 

3. y=sinx, Ay=sin (x + dx) — sin x 

= sin x cos dx + cos x sin dx— sin x. 
But cos dx = l+/i, sin dx=dx+I 2) (§16). 

.' . dy = cos x dx. 

4. To find the differential of the area A, Fig. 23. 

■ 

Y 





o x 

Fig. 23. Fig. 24. 

For the increment dx of x the increment dA of the area = PMNQ 
= PN + PRQ. PN=y dx, and PRQ is a part of RS, which =dx Ay 
and is /.a higher infinitesimal. 

.'. dA=y dx. 



43.] DIFFERENTIALS \s INFINITESIMALS 39 

For example, for the curve y x 8 , Fig. 24, dA x*dx } hence the 
relation connecting .1 and x is in this case .1 -{.r 1 . 

5. Barometric measurement of heights. Lei //,, be the height 

of a cubic inch o{ air at pressure p . Then the weight of a cubic 
inch at pressure p is (Boyle's law) u? p Po, ' l the temperature is 
tho same as before. Of a column of air of uniform temperature 

and one square inch in horizontal section consider the portion 
between sections at distances X, x + dx from the top, and let p, 
p+Jp be the pressures at top and bottom of this portion. Then 
Jp is the weight of this portion, dx its volume, its average pressure 
>p and <p + Jp and is therefore p + i, where i is infinitesimal. 
Hence 

w (p+i) j . w , p dp 

Jp = — - ax. . . dp= — pax. or dx = . 

po Po w o P 

This shows that the relation connecting x and p is 

x=— logp + c, (1) 

where c is a constant. Let the pressures at top and bottom of 
the whole column be P u P 2 , and h the total depth. Then p=P l 
when x=0, and p = P 2 when x=/i. Substituting in (1) and 
subtracting, 

ft = Pi(logP 2 -logP 1 )=^log(^). 

w w \jr, / 

The values of the constants p and w are to be supplied from 
experiment. 

M SCELLANEOUS EXAMPLES. 

dy y 2 (l-\ogx) 



1. x v ' =y x , show that 



dx x 2 {l—\ogy)* 



x-y 



2. llx=e v , dy/dx =\ogx/ (I +\ogx) 2 . 

/b + a cos x\ Va 2 — b 2 cfo 

4. d cos- 1 ( — ) = — — - 

\a-rocosav a + 6 cos a: 



40 INFINITESIMAL CALCULUS. [Ch. IX. 

5. Find the derivative of x with respect to tan x. 

Ans. cos 2 #. 



6. Find the derivative of sin -1 :r with respect to vl— x 2 . 

Ans. —x~ l . 

7. Differentiate tan x directly.* 

8. Differentiate tan _1 :r directly .f 

9. If f{x) =log (~£) , show that fix)+fiy) =/ (~^-\ . 

10. If (x—a) n is a factor of f(x), show that (x — a) n ~ l is a factor 
of f'(x). [Assume f(x) = (x —a) n F(x)]. 

Hence explain a method of finding the equal roots of an alge- 
braical equation. 

11. If f(x) contains a factor (x— a) -1 which causes it to be 
infinite when x = a, show that f'(x) is also infinite. 

12. A function f(x) is said to be an even function of its variable 
x if (f—x)=f(x), and an odd function if f(—x) = —f(x). What 
are the geometrical peculiarities of the graphs of such functions? 

Show that the following are even functions : 

cos#, x sin x, (e x —e- x )/x, x/(e x — l) + ix. 
Show that the following are odd functions : 

x 3 sec x, tanh x, log (sec z + tan x). 

13. Show that cosh -1 ^ and sech- 1 ^ are double-valued func- 
tions. 

* tan A— tan 5 = sin (A — B) /cos A cos B. 
t tan _1 m — tan— 1 w=tan -1 (m— n)/(l + mn). 



CHAPTER X. 
FUNCTIONS OF MORE THAN ONE VARIABLE. 

44. Such functions may be differentiated by the formulae 
already given. 

Ex. 1. u = (x + y 2 ) 3 . Here u is to be regarded as a function of x 
and y, both of which are assumed to have differentials. We 
have 

du=3(x + y 2 ) 2 d(x + y 2 ) =3(x + y 2 ) 2 (dx + 2y dy), 
,°. du=3(x + y 2 ) 2 dx + 6y(x + y 2 ) 2 dy. 



d 



C-) 



1 ( x \ 7 ^y 

2. w=sin _1 I — ), du = 



<y 







y 

ydx—xdy dx x dy 



y\/y 2 —x 2 \/y 2 —x 2 y\/y 2 —x 2 

45. Partial differentials and derivatives. It will be ob- 
served in these examples that the first term of the result is 
what we should have obtained if we had differentiated u on 
the supposition that x alone varied , y being regarded as a 
constant; let this be written d x u. The second term is what 
we should have obtained if we had differentiated on the 
supposition that y alone varied, and this we call d y u. Hence 
in these examples 

du~d x u+d y u. 

41 



42 INFINITESIMAL CALCULUS. [Ch. X, 

The same thing is true for all continuous functions of two 
variables. For, if we differentiate by the ordinary methods, 
we shall in every case get a result which may be written 

du = M dx + N dy, 

where M and N may contain x and y, but not dx or dy. If 
dy = 0, the right-hand side reduces to M dx, which is therefore 
d x u, the differential of u on the supposition that y is constant 
(i.e., dy=0) while x varies. Similarly N dy is d y u, the differ- 
ential of u on the supposition that y alone varies. 

.'. du = d x u + d y u. (1) 

The differentials d x u, d y u are called partial differentials 
of u with regard to x or y, du being called the total differ- 
ential of u. The total differential is therefore equal to the 
sum of the partial differentials. 

Similarly if u is a function of three variables x, y, z, 

du == d x u + d y u + d z u. (2) 

It should be noticed that formulae (C), (Ci), (D), (G 2 ) are 
particular cases of functions of two or more variables. 
The result (1) may be put into the form 

7 a x u 7 ayU n 

du= ^ dx+ __ dy> (3) 

which brings out clearly the fact that the coefficients of 
dx and dy are the partial derivatives of u, which are equal 
to the partial differential quotients. The subscripts are 
usually omitted, (3) becoming 

7 du 1 du _ 
au = — ax-\-— dy. (4) 

The symbol d is frequently employed to express partial 
differential quotients or derivatives, (4) being written * 

, du , da , 
du = — <±c + — dy. (5) 



* du/dx may be read "partial du by dx." 



46.] FUNCTIONS OF MORE THAN ONE VARIABLE. 43 

Ex. 1. w=sin {x 2 + xy). Differentiating, first regarding x only 
as variable, and afterwards regarding y as the variable, 

du/dx =cos {x 2 + xy) . (2x + y), du/dy = cos (x 2 + xy) x. 
2. u = x 3 + y 3 + z 3 — 3xyz, 

du/dx =3(x 2 — yz), du/dy =3(y 2 —zx), du/dz=3(z 2 — xy). 

46. If u = f(x, y), dx and dy, being differentials of the 
variables, are increments of x and y. If dx and dy are taken 
as infinitesimal increments, du is not the same as Ju, the 
infinitesimal increment of the function. Since it may be 
obtained by the ordinary rules of differentiating, du is Ju 
when the higher infinitesimals are omitted (§ 42), or 

Ju=du+I, 

Hence when dx and dy are very small du is a close approxi- 
mation to Ju. 

Examples. 

1. w=sin (x 2 — y 2 ), dxu =2x cos (x 2 — y 2 ) dx, 

d y u = —2y cos (x 2 —y 2 ) dy. 

2. u = (x—y)/(x + y), du = 2(y dx— x dy)/(x + y) 2 . 

3. u = (ax 2 + by 2 + cz 2 ) n , 

n-l 



du=2nu n (ax dx + by dy +cz dz). 

4. If tan 6 =y /x, (x 2 + y 2 ) dd =x dy —y dx. 

5. u=x y , d x u=y x v ~ x dx, d v u = (log x) x v dy. 

: . du=y x v ~ l dx+ (log x) xv dy, as in (G 2 ). 

6. u=log(e? + eV), du/dx + du/dy = l. 

7. u=t&n- 1 (x/y), du/dx = y/(x 2 + y 2 ), d u /dy = — x/(x 2 + y 2 ). 

8. u =102: (tan x + tan y). sin 2a:— + sin 2 v— =2. 

& * ' dx *dy 

9. u=\og y X, UX du/dx + y du/dy =0. 
[Note, logy x = log, x /logo y.] 

10. Given x =r cos 0, y =r sin 0, show that 

dx 2 + dy 2 = dr 2 + r 2 d0 2 , 
x dy —y dx =r 2 dQ. 



44 INFINITESIMAL CALCULUS. [Ch. X. 

11. (1) If a function u consists of terms such as axPyQ, and 
p + q is the same number n in each of the terms, u is said to be 
homogeneous and of the degree n. Show that for such a function * 

du du 
x — \-y — =nu. 
dx oy 

(2) If u=f(v) and v is homogeneous as defined, show that 

du du „.- 

x— + y—=nvf (y). 
dx dy 

These propositions may obviously be extended to functions of 
three or more variables. Verify in the case of Exs. 1 and 3. 

47. Tangent and normal. Let f(x, y) = c (c constant) be 
the equation of a curve. The. first member of the equation 
is a function of x and y; calling it u and differentiating the 
equation we have, by § 45 (5), 

¥/ X + ¥y dy=0 > - (1) 

whence dy/dx, the slope of the tangent at (x, y) } is — — / — • 

Let (xi, y\) be the point of contact of a tangent to 
the curve, (x, y) any other point on the tangent. Then 
x — x\ and y — yi are proportional to dx and dy. Hence, 
from (1), 

(!),<— > + (l), ( s<-^ (2 > 



is the equation of the tangent, and 

x — x^ y — yi 



'?)u\ /du\ 
fix) 1 \dy)i 



(3) 



the equation of the normal, at (x\, y\). These equations 
are often more convenient than those of § 33. 

* A particular case of Euler's theorem on homogeneous functions 
(Ex. 1, §234). 



•17. is.] FUNCTIONS OP MORE THAN ONE VARIABLE. 45 

Ex. Find the equations of the tangent and normal at the 
point (a, a) on the curve 

x 3 -f y 3 — 2a xy=0. 

du /dx = 3x 2 -2ay =3a 3 -2a 2 - a 2 for (a, a). 

du /dy =3y 2 -2ax =3a 2 -2a 2 =a 2 for (a, a). 

:. the tangent is a 2 (x — a)-\-a 2 (y — a) = 0, or x + y=2a, 

. . . . x —a y —a 

and the normal is — -- = — — , or x=y. 

a 2 a 2 

48. Centre of a conic. Let the general equation of a conic 
be 

ax 2 + 2hxy + by 2 + 2gx + 2fy + c = 0, 

or a = 0. When referred to parallel axes through the point 
(xi, 2/1) the terms of the first degree are 

2(axi +hy 1 +g)x + 2(hxi+byi+f)y, 

/du\ /du\ 

The new origin is therefore the centre if ( — ) =0 and 

\ ox) 1 

(^— ) =0. Hence the centre of the conic is the intersection 

of 'du/dx = and du/dy = 0. If the coordinates of the point 
thus found satisfy the given equation, the centre is on the 
conic, which therefore consists of a pair of straight lines. 

Examples. 

1. Find the equation of the tangent at any point of the curve 
(x/a) m + (y/b) m =2, and show that x/a + y/b=2 is the tangent at 
the point (a, 6). 

2. Show that the length of the perpendicular from the origin 
on the tangent at (x, y) to the curve u=c is 



du du\ I ,du\ 2 /dio 

+ 



/ du du\ I 



y/ 1 N \°xl \dy 



46 INFINITESIMAL CALCULUS. [Ch. X. 

3. In the case of the curve x% + y5=a%, Fig. 18, show that this 

perpendicular =^/axy. 

4. In the case of the parabola (x/a)% + (y/b)*=l, show that 
this perpendicular =[abxy/(ax + by]*. 

5. Find the centres of the conies 

(1) x 2 -4xy-2y 2 + 6y=2, Arts. (1, £). 

(2) 18x 2 -8xy + 3y 2 + 8x-6y-5=0. (0,1). 

6. Show that 3x 2 + 5xy—2y 2 —x + 5y=2 represents a pair of 
straight lines. 



CHAPTER XI. 

SMALL DIFFERENCES 

49. When the differentials of the variables of a function 
are small increments, the differential of the function is a 
close approximation to the increment of the function (§§ 42, 
46). 

Examples. 

1. Given sin 30° =i cos 30° =£^3", find sin 30° 1'. 

Here the angle increases by a small amount and it is required 
to find the small increment in the sine. 

We have d sin d = cos dO; cos 0=£\/3, dO =60/206265 rdn., 
.'. dsin ='0002519, .". sin 30° V ='5002519, which is correct to 
the last decimal place. 

2. How much must be added to log l0 sin 30° to get log 10 sin 30° 1'? 

We have d log sin =cos dd/s'm 6 ='0005038, which is the in- 
crease of the Napierian log. ; the increase of the common log. is 
obtained by multiplying by the modulus. 

.'. '0005038 X* 4342945 ='0002188 

is the required increment. 

The difference columns in the mathematical tables are found 
or verified in this way. 

3. The radius of a right circular cone is 3 inches and the height 
is 4 inches; if the radius were '006 in. more, and the height '003 in. 
less, what would be the change in the volume? 

The volume v=%7zr 2 h, .'. dv=\-(2rh dr + r 2 dh) 

= Jtt(2 X3 X4 X'006 -3 2 X'003) =' 1225 cub . in. 

4. Assuming that the radius of an iron ball increases by '000011 
of its original length for each degree of temperature, what will 

47 



48 INFINITESIMAL CALCULUS. [Ch. XT. 

be the increase in volume of an iron ball of 8 in. radius when the 
temperature is raised 25 degrees? 
The volume v =f 7rr 3 , .* . dv =47rr 2 dr 

=4ttX8 2 x25x*000011x8 = 1.77 cub. in. 

5. In a certain triangle, b =445, c =606, A =62° 51' 33", whence 
a is calculated and found to be 565; it is then noticed that A 
should have been 62° 53' 31"; what is the correction to a? 

The change in A is V 58" = 118" = 118/206265 rdn. 
Also, a 2 = b 2 + c 2 —26c cos A ; differentiating this, supposing b 
and c constant, we have 

2a da=2bc sin A dA, .'. da=bc sin A dA/a 
= 445 X 606 X sin 62° 51' 33" X 118/206265x565 =243. 

An approximate value of sin A is sufficient in this place. 

6. Given loge 900=6*8024, find log e 901. 

Increase of log x =dx/x = l/900. Arts, 6'8035. 

7. Given log 10 1000 =3, find log 10 1002. Arts. 3*00087. 

8. Find tan 45° 1'. Arts. 1*00058. 

9. On account of the rotation of the earth the correction to 
the weight w of a body is — w cos 2 ^/289, where ^ is the latitude. 
What is the change in this correction for one mile north of lati- 
tude 45° N.? the radius of the earth being assumed to be 4000 
miles. Ans. w/(289x4000). 

10. Find the relation connecting small differences of t and d in 
the equation 

sin h=sin <f> sin d + cos <j> cos d cos t, 

</) and h being constant. 
Differentiating and arranging the terms, we get 

/tan <j> tan d\ 

dt=[-7~-, )dd. 

\ sin t tan t' 

This is the ''Equation of Equal Altitudes' ' in astronomy. 

11. Find the relation connecting small differences of d and A 
in the equation 

sin d =sin <£ sin h — cos <j> cos h cos A f 

<f> and h being constant. 

Ans, dA =cos d d<V(cos <£ cos h sin A). 



50.] 



SMALL DIFFERENCES. 



49 



12. In any plane triangle 

(1 ) da = cos C db + cos B dc + b sin C dA, 
da db dA dB 
a b tan A tan B' 

13. The sides a and b of a right-angled triangle ABC (C=90°) 
receive small corrections da and d6; what is the change in the 
perpendicular p from the right angle on the hypothenuse? 

Arts. -, ad *-r+ -, or dp = cos 3 A dA + cos 3 B db. 
p 3 a 3 b 3 

50. Solution of equations by approximation. If a is a 

value of x, and h & small increment of x, then, § 42 (2), 

f(a + h) — f(a) = f'(a)h, nearly. 

Let }(x) = be an equation, a a quantity which is known 
(by trial or otherwise) to be an approximate value of a 
root of the equation, a + h to be that root, where h is small 
compared with a. Then f(a + h) = 0. .'. h= —f(a)/f'(a), 
nearly. Hence if a be a first approximation to the root, 

/(a) 



a 



/'(a) 



(1) 



will be a nearer approximation. If this new value be sub- 
stituted for a in (1), a nearer approximation still will be 
obtained, and with this a closer approximation, and so on. 




Fig. 25. 



If Fig. 25 is the graph of f(x), the roots of the equation 
f(x) = are the intercepts of the curve on the z-axis. If 
OA is the true value of a root and OB the assumed value, 
the first corrected value is OC. For, if OB=a, j{a) = BE 7 



50 INFINITESIMAL CALCULUS [Ch. XI 

/'(a) = tan BCE, .*. f(a)/f{a) = CB. If OC is next assumed, 
the second corrected value is OD. 

Examples. 

1. x 3 -4x -2=0. Here f(x) = x*-4x-2, and f'(x) = 3z 2 -4. 
Since /(2) = — 2, and /(3) = 13, there is a root of the equation 
lying between 2 and 3. Let us then assume 2 as the first approxi- 
mation. Then 

/(2) -2 

= 2 — —=2*25 



/ / (2) 8 

is a second approximation. Again, 

2*25 - {) 2 ^\ =2 * 2 5 -'035 =2*215, 

/'(2*25) ' 

a third approximation. 

Since /(0) = — 2, and /(— 1) = 1, there is a root lying between 

and —1. Assume it to be —'5. Then the next approximation 

is — *538. Again, taking — *54 as this second approximation we 
find the next to be —'5392. 

Similarly x = —1*676 is the third root. 

2. x 3 + 2z-13=0. Ans. 2*069. 

3. x 3 -x 2 -2=0. 1*696. 

4. x* =34. 2*024. 

5. x* -12^=200. 2*982. 

6. e x {\ + £ 2 ) =40. 2*046. 

7. x*=5. 2*129. 

8. z 4 -12z 2 + 12:c-3=0. 2*858, -3*907 



CHAPTER XII. 



MULTIPLE POINTS. 



Si. Tangents at the origin. Let it be required to find 
the line which touches the curve x s + y s = 3axy, Fig. 28, 
at the origin. 

Differentiating the equation we obtain 

dy/dx = — (x 2 — ay) I (y 2 — ax) , 

which for the point (0, 0) assumes the form 0/0. The 
difficulty here met with is avoided by the method now to 
be explained. 

52. If a curve passes through the origin, its equation can 
contain no constant term; let it be 

aix + biy + a 2 x 2 + b 2 xy + c 2 y 2 + . . . = 0. 

For x and y substitute r cos d and 
r sin d, where r is the length of a 
straight line drawn from the origin to 
the point (x, y) on the curve, and 
i? the angle which this line makes 
with the x-axis. The equation be- 
comes 




Fig. 26. 



r(ai cos + bi sin 6)+r 2 (a 2 cos 2 # + . ..) + .. . = 0. 

One root is r = 0, which implies that the curve passes through 
the origin; the remaining roots are given by the equation 



a x cos 6 + b\ sin + r(a 2 cos 2 9+ ...) + ••• =0. 



51 



52 INFINITESIMAL CALCULUS. [Ch.XII. 

Another root will = if «i cos d + bi sin d = 0. Hence if 
the tangent at the origin makes an angle <fi with the x-axis, 
(j> is given by the equation 

a\ cos <j> + b\ sin cf> = 0. (1) 

If (x, y) is a point on the tangent at a distance t from 
the origin, cos cf) = x/t and sin $ = y/t. Substituting in (1) 
we have a\x + biy=Q for the equation of the tangent at the 
origin, i.e., the terms of the first degree in the given equa- 
tion, equated to zero, represent the tangent at the origin. 

If there are no terms of the first degree, it may be shown 
in the same way that a 2 x 2 + b 2 xy + c 2 y 2 = is the equation 
of a pair of tangents at the origin; and generally, yjhen the 
origin is a point on the curve, the terms of the lowest degree, 
equated to zero, represent the tangents at the origin. 

53. Multiple points. A point at which there are two or 
more tangents (i.e., where two or more branches of a curve 
intersect) is called a multiple point; it is called a double 
point, a triple point, etc., according as two, three, etc., 
branches intersect at the point. 

When the equation a 2 x 2j rb 2 xy + c 2 y 2 = represents a pair 
of distinct lines the point is called a node (Figs. 27, 28). 

When the lines are coincident the two branches of the 
curve touch one another and the tangent may be considered 
as a double tangent. Such a point is called a cusp, w 7 hich 
is said to be of the first or second species according as the 
two branches of the curve lie on opposite sides (Figs. 29, 30) 
or on the same side (Fig. 31) of their common tangent; and 
to be double or single according as the branches lie on both 
sides (Fig. 34) or on one side only (Fig. 29) of their common 
normal. A cusp is also called a stationary point; for, con- 
sidering the curve as the path of a moving point, at a cusp 
the point must come to rest and reverse its motion. 

When the lines are imaginary the point is called a con- 
jugate point. The coordinates of such a point satisfy th§ 



53.] 



MULTIPLE POINTS. 



53 



equation of the curve, but the point is isolated from the rest 

of the locus which the equation represents. 

Examples. 

1. The lemniscate* a 2 (y 2 -x 2 ) + (y 2 + x 2 ) 2 =0, Fig. 27. 
The origin is a node at which the tangents are y 2 0, i.e., 

y = x and y = — x. 

.A 





Fig. 27. Fig. 28. 

2. The folium t x 3 + y 3 =3axy, Fig. 28. 

The origin is a node, the tangents being given by xy=0, i.e., 
2=0, y=0, the axes. 

3. The semi-cubical parabola ay 2 =x 3 , Fig. 29. 

The origin is a cusp, the tangents being given by y 2 =0, i.e., 
two lines coinciding with the axis of x. Moreover, the curve is 
symmetrical with regard to the axis of x } and y is impossible if x 
is negative; hence the cusp is single and of the first species. 

4. In the curve (y—x) 2 =x 3 , Fig. 30, the origin is a cusp at 
which the tangent is y=x; also, since y=x±x%, y> x on one 
branch and <x on the other, hence the cusp is of the first species. 

* The curve is most easily plotted from its polar equation 

r 2 = a 2 cos 20. 

t This curve may be plotted as follows: Let y = mx in the equation. 
Then x = ?>am/(\-{-m 3 ) and y = mx or Sam 2 /(\ -f m 3 ). 

Thus x and y are expressed in terms of a third variable, and by giving 
arbitrary values to m the coordinates of any number of points on the 
curve may be calculated. The same substitution may be employed 
in other cases (e.g., Figs. 36, 37, 38) in which the equation contains 
terms of two degrees only. It should be noticed that /// is the slope 
of the line drawn from the origin to the point (x, y) on the curve. 



54 



INFINITESIMAL CALCULUS. 



[Ch. XII. 



5. In the curve (y—x 2 ) 2 =x*, or y=x 2 (l±Vx), Fig. 31, the 
origin is a cusp, the tangent at which is y=0; also, y is + on 
both branches until x = 1, and / . the cusp is of the second species. 






Fig. 29. 



Fig. 30. 



Fig. 31. 



6. In the curve y 2 =x 2 (2x + l), Fig. 32, the origin is a node 
at which the tangents are y=±x. But in the curve y 2 =x 2 (2x — l), 
Fig. 33, the tangents are y 2 =—x 2 , and are .*. imaginary, and 
hence the origin is a conjugate point. 






Fig. 32. 



Fig. 33. 



Fig. 34. 




Fig. 35. 



7 There are certain cases in which the origin is a conjugate 
point even when the terms of the second degree are a perfect 
square. Thus, in the curves y 2 = x 4 (2x + l), Fig. 34, and 
y 2 =x 4 (2x — 1), Fig. 35, the origin is a double point and the tan- 
gents are given by y 2 =0; in the first curve the origin is a double 
cusp, in the second a conjugate point, since y is imaginary for 
any value of x less than \. 

8. The curve ay 3 —3ax 2 y =x 4 , Fig. 36. 

The origin is a triple point at which the tangents are 
ay 3 — 3ax 2 y =0; i.e., y =0, y = ±x\ / S. 

9. In the curve ay 4 — ax 2 y 2 = x : \ Fig. 37, the origin is a quad- 
ruple point, at which the tangents are y=0 y y=0,y = ±x. 

10. (x— y) 2 = (x — 1)\ The point (1, 1) is a cusp, for the equa- 
tion referred to parallel axes through (1, 1) is (x— y) 2 =x*. 



54.] 



MULTIPLE POINTS. 



55 



11. Find the tangents to the following curves at the origin: 

(1) (a 2 -\-x 2 )y 2 = {a 2 -x 2 )x 2 . Ans. y=±x. 

(2) a 2 y(x + y)=x 4 . y =0, x + y = 0. 

(3) x{y-x) 2 =y\ x =0, y=x, y=x. 

(4) a(y-x)(y 2 + x 2 ) + x 4 =0. y=x. 

(5) y 3 (y-x)=a(y 3 + x' d ). x + y=0. 

(6) (x-a)y=x(x-2a). y=2x. 

(7) y 2 =(x-l)x 2 . Imaginary. 





Fig. 36. 



Fig. 37. 



12. Show that the origin is a single cusp of the first species on 
the cissoid y 2 (a—x)=x 3 , Fig. 41. 

13. Show that there is a node at the point (1, 2), on the curve 
(y-2) 2 = (x-l) 2 x. 

14. Show that the point (2a, 0) is a node on the curve ay 2 = 
(z-a)(z-2a) 2 . 

15. Show that the point (—a, 0) is a conjugate point on the 
curve ay 2 =x{a + x) 2 . 

54. Let the equation of a curve, freed (if necessary) from 
fractions and radicals affecting the coordinates, be f(x, y)=c 
or u = c. The tangent, § 47 (2), when referred to parallel 
axes through the point of contact {x\, y{) is 



( 



cm\ (du\ n 



56 INFINITESIMAL CALCULUS. [Ch. XII. 

Hence if (— ) =0 and (^-) = the equation of the 
Xdx/i \dy/i 

curve referred to the new axes will have no terms of the 

first degree. Conversely, points whose coordinates satisfy 

du/dx = and du/dy = as well as the given equation 

u = c are multiple points of the curve. 

Ex. 1. To examine the curve (x — l) 5 — (2x — y) 2 =0 for multiple 
points. 

du/dx =5(x-l) 4 -4(2a: -y) =0, 
du/dy =2(2x -y) =0, 

whence x = l } y=2. These coordinates satisfy the given equa- 
tion, hence (1, 2) is a multiple point. Transforming to parallel 
axes through (1, 2) the equation becomes x 5 — (2x— y) 2 =0, hence 
the point is a cusp at which the tangent is 2x ==y. 

2. Examine the curve x 3 — 2x 2 + y 2 — 4x-f 2y-\-§=0 for multiple 
points. Arts. A conjugate point at (2, —1). 



CHAPTER XIII. 
ASYMPTOTES. 

55. Definition. An asymptote of a curve is the limit of 
position of a secant when two of its points of intersection 
with the curve move away to an infinite distance, and hence 
also the limit of position of a tangent when the point of 
contact moves to an infinite distance. 

56. Asymptotes by substitution. 

Ex. 1. Of the curve x 3 + y 3 =3axy, Fig. 28, the line y=mx + b 
is an asymptote if m and b are determined so that the line may 
meet the curve in two points infinitely distant. Substituting 
mx + b for y in the equation of the curve we have 

(l+m 3 )# 3 + 3(ra 2 6— am)x 2 + . . . =0, (1) 

the roots of which are the abscissas of the points of intersection 
of the line and the curve. Two of the roots become infinite * 
when m and b change so as to cause the coefficients of the two 
highest powers in (1) to =0. Hence the required values of m 
and b are obtained by solving the equations 

l+m 3 =0, m 2 b—am=0. 

.'.m=—l and b=—a. Hence the asymptote is y=—x—a, or 
x + y + a=0. The result might have been obtained equally well 
by the substitution x=my + b. 

* The equation a x n + a x x n - l -\- . . . J ra n - l x-\-a n = (2) 

is obtained from a n x n -\-a n - l x n - l -\- . . .-\-a i x-\-a = (3) 

by changing x into 1/x. The roots of (2) are the reciprocals of those 
of (3). Hence if a and a x change and = 0, two roots of (3) = and 
• *. two roots of (2) become infinite. 

57 



58 INFINITESIMAL CALCULUS. [Ch. XIII 

2. Find the asymptotes of the hyperbola ——^- = 1. 

a h 

Ans. y= ±—x. 

a 

57. Asymptotes by expansion. The following definition of 
an asymptote gives a better idea of the relation of the line 
to the curve: 

Def. When the distance (measured parallel to an axis) 
between a line and a curve is infinitesimal as both recede 
to an infinite distance, the line is said to be an asymptote 
to the curve. Such lines may be rectilinear or curvilinear. 
If the equation of a curve, when y is expressed as a series 
of descending powers of x, take the form 

c d 

y = ax + b-\ \--z + . . . , (1) 

x x z 

the line y=ax + b will be a rectilinear asymptote. For the 
difference between the y of the curve and the y of the line 
is c/x + d/x 2 + . . . , which is infinitesimal when x is infinite. 

The line y = ax + b is also the limit of a tangent of the 
curve (1). For the slope of the tangent = dy/dx = a — cx~ 2 
— . . . =a for x infinite, and the ^/-intercept of the tangent 
= y—x dy/dx = b + 2cx~ 1 + . . . =b. 

The sign of the term c/x in (1) will determine whether the 
curve lies above or below the asymptote when x is very 
large. 

If the equation take the form 

y=ax 2 + bx + c-\ h-« + . . . , 

xx 1 

there will be a curvilinear asymptote, viz., the parabola 
y = ax 2 + bx-\-c. 

Ex. 1. i/ 3 =x 3 + 3ax 2 , Fig. 38. 

/ 3a\ / 3a\ * . . . . 

We have y 3 =x 3 il-\ ), or y=xil-\ — I , which by the 



57.] 



ASYMPTOTES. 



59 



binomial theorem (when a;|>|3a) 






a* 

or y=x + a h...; 

x 



.*. y=x + a is an asymptote. The curve lies below the asymptote 

when a: is a large positive number, and above it when a; is a large 

negative number. 

x 2 ii 2 bx / x 2 \ * 

2. The hyperbola — -f- = 1 . Here y=±—(l — -) 
JC a 2 b 2 a \ a 2 / 

bx I a 2 \ bx ab 

bx 
.*. the asymptotes are y=±—' 





Fig. 38. 



3. fy(x-l)=x 3 , Fig. 39. 

x 3 



By division, 4y = 



x 2 + x + l + 



Fig. 39. 



1 



x — 1 ' x — 1" 

When x is very large the last term is very small and =0, and 
the ordinate of the curve = that of the parabola 4y=x 2 + x + l, 
which is called a parabolic asymptote. (The line AB is the axis 
of the parabola.) It will be noticed that this curve is asymptotic 
to the given curve both when x is + and when x is — . The line 
x = l is a rectilinear asymptote, as will be seen from § 58. 






60 



INFINITESIMAL CALCULUS, 



[Ch. XIIL 



58. Asymptotes parallel to the axes. Let the algebraical 
equation of a curve, freed (if necessary) from fractions and 
radicals affecting the coordinates, and arranged in descending 
powers of x, be 

fi(y)x™ + J 2 (y)x m - 1 + f 3 (y)x m - 2 + . . . = 0, 



whence 



/i(2/)+/ 2 (^+/ 3 (2/)4+...=0. 



(1) 



If there is an asymptote parallel to the #-axis, y remains 
finite when x is infinite. Hence all the terms of (1) after 
the first become infinitesimal, and the y of the curve ap- 
proaches a limit which satisfies /i(j/) = 0, i.e., 3/ = the y of 
a line y—a = Q, y~a being a factor of fi(y). Hence, when 
the equation of a curve is arranged according to powers of 
x, the coefficient of the highest power, equated to zero, 
represents the asymptotes which are parallel to the z-axis. 
The asymptotes parallel to the y-axis may be found in the 
same way. 





Y 




i 




— 




~~~— -^ 


/ 






^_^-^-^c 


\ 




X 


. 




\ 








1 


1 






Fig. 40. 



Fig. 41. 



Ex. 1. x 2 y 2 -3xy 2 -x 2 + 2y 2 =0, Fig. 40. 

Arranged according to powers of x the equation is 

(y 2 -l)x 2 -3y 2 x + 2y 2 =0, 



58, 59.] ASYMPTOTES. 61 

and according to powers of y } 

(x-l)(x-2)y 2 -x 2 =0. 

Hence y = ±1, and x = l, x=2 y are asymptotes parallel to the 
axes. 

2. The cissoid y 2 (a—x)=x 3 , Fig. 41. 

The line a —x = 0, or x = a is an asymptote parallel to the ?/-axis. 

59. In any equation the terms of the highest degree, equated 
to zero, represent lines drawn through the origin. The 
equation which gives the slopes of these lines is found by 
substituting mx for y, or m for y/x. This is the same equa- 
tion as that which determines the slopes of the asymptotes 
(§ 56). Hence the terms of the highest degree, equated to 
zero, represent lines drawn through the origin in the direc- 
tion of the infinite branches of the curve. 

Examples. 

1. x 3 —y 3 =3axy. Arts. y=x—a. 

2. y 3 =x 2 y + 2x 2 . y= ±'x + l, y + 2 = 0. 

3. x 4 =xy 3 + 3y 3 . x-y = l, z + 3=0. 

4. x 2 y = x 3 -\- x -{- y . y=x,x=±l. 

5. x 4 — y 4 + x 2 =4txy 2 . x±y = \. 

6. x'=x 2 y 3 -(l-x)y 3 . 3(x-y) = l, 2x~ ±VK-1. 

7. axy=x 3 —a 3 . x=0 y ay=x 2 . 

8. x 3 + y*=a 3 . x + y=0. 

9. x 3 -27y*=2x 2 . 3x-9y-2=0. 

10. y + xy=x 3 . x + l = 0, y =x 2 — x + 1. 

11. y =tan x. The y of the curve =00 when the x = \n y : . the 
line x=\tz is an asymptote (§ 57). Similarly x = (n + i)n, n any 
integer, is an asymptote. The same lines are asymptotes to 
y=sec x. 

12. Show that y = l and y= — 1 are asymptotes of ?/=tanh x. 

13. Show that the curve 2y=e x is asymptotic to 2/=sinh x, 
y = cosh x, and y=Q, 



CHAPTER XIV. 



TANGENT PLANES. TANGENTS TO CURVES IN SPACE. 



60. Geometrical illustration of partial and total differen- 
tials. Let z = f(x, y). Values of x and y determine z, and 
hence a point (x, y, z) in space referred to axes which we 
shall assume to be rectangular. Points thus obtained lie 
on a surface which is the locus of the equation z=f(x, y). 
This surface is a geometrical representation of the function. 



z 

F 


i\^^^Z-^ 


0/ 


K 

J 

1 


H 
G 

E 






R / 
F 


1 













/" 




H* 






M 




r N 







C 


» 



Fig. 42. 

Let OA = x, AM=y, and MP = z. Then P is a point on 
the surface. Let OB=x + dx, BC=y+dy, and let the new 
value of z be CQ. Then Q is another point on the surface. 
The plane PF parallel to XOY cuts off CF = MP, hence FQ 
is Az y the increment of z. Planes through P and Q parallel 
to XOZ and ZOY cut the surface in PH, HQ, QK, KP. 

62 



60-62.] . TANGENT PLANES. 63 

Draw PJ and PG tangents to PK, PH, and let PGRJ be 
the plane through PJ, PG. If we suppose y to be constant, 
we are confined to the plane PN (produced if necessary); 
Pl = dx and PJ touches PK, hence I J is d x z. Similarly 
EG is d u z. Let a line through the middle point of MC parallel 
to OZ meet the plane PGRJ. This lme=$(MP+CR) in 
the trapezium PMCR, and also =%(DG + NJ) in the trape- 
zium GDNJ. .'. FR = IJ + EG = d x z + d y z. But dz = d x z + 
d y z (§ 45). Hence FR is dz. The tangents of the angles 
IPJ, EPG are the partial derivatives of z with respect to 
x and y, i.e., they are dz/dx and dz/dy. 

6i. Tangent plane. When Jz and dz are infinitesimal the 
latter is the part of the former, which contains the infini- 
tesimals of the lowest order (§ 46). Thus FQ and FR 
correspond in the plane PMCQ to Ay and dy of § 42. Hence 
the straight line PR touches the section of the surface made 
by the plane PMCQ, and therefore the plane PGRJ is the 
locus of all such tangent lines at P, for dx and dy are any 
increments. Such a plane is defined to be the tangent plane 
at P. 

Notice that if {x, y, z) is the point of contact, and dx, dy 
are any increments, {x + dx, y + dy, z + dz) is any other point 
in the tangent plane. 

62. Equation of the tangent plane. Let the equation of 
the surface be f(x, y, z) = c or u = c. Differentiating, 

?)u , , du. , du . 

— d x + 7T- d y + — dz = . 
ox oy oz 

Let (x\, y\, z{) be the coordinates of the point of contact 
P, Fig. 42, (x, y, z) those of any other point in PR and there- 
fore of any other point in the tangent plane. Then x — x\, 
y — yi, z — Zi are proportional to dx, dy, dz. 

is the equation of the tangent plane at (x\, y%, z\). 



~ 



64 INFINITESIMAL CALCULUS. [Ch. XIV. 

Ex. To find the tangent plane at the point ( — 1, 1, 2) on the 
surface x 3 -x 2 y + y 2 + z = l. 

du/dx =3x 2 —2xy =5 for the point (— 1, 1, 2), 
du/dy= —x 2 + 2y = l for the point ( — 1, 1, 2), 
du/dz = l. 

Hence the tangent plane is 

5(x + l) + (2/-l) + (3-2)=0, or 5x + y+z + 2=0, 

63. Equations of the normal. The normal passes through 
(x\, 3/1, z\) and is perpendicular to the tangent plane; hence 
its equations are 

x-xi y-yi z-z x 

(du\ fdu\ /du\ ' w 

\dx/ 1 \dy)i Vdz/i 

64. Tangent plane at the origin. Conical points. Let the 

equation of the surface be freed (if necessary) from fractions 
and radicals affecting the coordinates. If the origin is on 
the surface the equation will contain, no constant terms, 
and by substituting r cos a, r cos /?, r cos y for x. y, z, it 
may be shown exactly as in § 52 that the terms of the first 
degree, equated to zero, represent the tangent plane at the 
origin. Similarly, if there are no terms of the first degree, 
those of the lowest degree present will represent a surface 
touching the given surface at the origin. This tangent 
surface is generally a cone, in which case the origin is called 
a conical point; but it may be two or more planes.* 

As in § 54, it may be shown that the coordinates of a 
conical point or a point where there are two or more tan- 
gent planes will satisfy du/dx = 0, u/dy = 0, du/dz = 0, as 
well as the given equation u = c. 

* A homogeneous equation with no constant term represents a 
locus of straight lines passing through the origin. For, if satisfied by 
x, y, z, it is satisfied by ex, cy, cz, the coordinates of any other point 
on the line joining the origin to (x, y, z). 



63-60.] 



TANGENT PLANES. 



65 



Ex. 1. Of the surface x 2 — y 2 — z 2 + x 3 =0, x 2 —y 2 —z 2 =0 is a tan- 
gent cone at the origin,, 

2. Of the surface x 2 — y 2 — z 2 — 2yz =x 3 , x + y + z=0 and x—y—z 
= are tangent planes at the origin. 

3. Find a conical point on the surface x 3 + y 2 + 2yz — 3x— 42=2. 

Arts. (1, 2, -2). 

65. Centre of a quadric. Exactly as in § 48 it may be 
shown that the centre of any surface whose equation u = c 
is of the second degree is obtained by solving the simulta- 
neous equations ?>u/dx = 0, du/dy = 0, du/2)Z = 0. 



Ex. Find the centre of x 2 



3y 2 -z 2 + 4yz-4:X + 8y-6z=0. 

Ans. (2, 2, 1). 



66. Curve in space. Let P and Q be two points near one 
another on a curve, P being 
(x, y, z) and Q (x + dx, y + dy, 
z + dz). Then dx = AB=CE = 
PG, dy = ED = GF, and dz = FQ. 
Let the arc PQ = ds and the 
chord PQ = q. The tangent- PT 
is the limit of position of the 
secant PQ when Q approaches 
coincidence with P. Let a,/?, y 
be the direction angles of PT. 
Then a = HPT, and 

cos a = £ cos GPQ = £(dx/q) 
= £(dx/ds) (§ 17) =dx/d's. 

Similarly cos /? = dy/ds, cos y = dz/ds. 

Squaring and adding, 




Fig. 43. 



'dx\ 2 (dy\ 
<ds/ \ds/ 



+ 



(dz_y 

\ds) : 



or ds 2 = dx 2 +dy +dz 2 . (1) 



Draw TK parallel to ZO to meet the plane PFG, and KH 
parallel to YO. Then if dx is PH, dy is HK, dz is KT, and 
ds is PT. 



66 INFINITESIMAL CALCULUS. [Ch. XIV. 

67. If the coordinates of a point on a curve are given in 
terms of a fourth variable, dx, dy, and dz may be written 
down at once. Usually, however, a curve in space is given 
as the intersection of two surfaces. Let the surfaces be 
u = C\ and v = c 2 . Differentiating, 

^r-dx + ^r dy+—dz = 0, (1) 

ox oy dz 

and 7t- dx + — dy + —dz = 0. (2) 

ox dy dz 

If (x, y, z) is the point of contact P of a tangent plane, 
(x + dx, y + dy, z + dz) is any other point Q in the plane. 
Hence if P(x, y, z) is a point in the curve of intersection of 
the surfaces, and (1) and (2) are simultaneous in dx, dy, dz, 
Q(x J rdx, y + dy, z + dz) is any other point in the line of inter- 
section of the tangent planes, and PQ is the tangent at P 
to the curve of intersection. 

The plane which is perpendicular to the tangent line* at 
the point of contact is called the normal plane of the curve. 

Ex. Find the equations of the tangent to the curve 

x 2 -2y 2 + ?/2 + 3=0, xy-z 2 + x + 4=0 

at the point (1, —1, 2). 

Differentiating the equations, 

2x dx + (2 —4y)dy -\-y dz=0, 
(y + 1 )dx + x dy—2z dz=0, 

or, for the point (1, — 1, 2), 

2dx + 6dy— dz=0, dy—4:dz=0, 
. dx dy dz 

whence — = T = T 

Since dx, dy, dz are proportional to the direction cosines of the 
tangent, the equations of the tangent are 

x-l_y+l _z-2 

~^23 8 2" 

The normal plane at (1, —1, 2) is 

-23(3-l) + 8(t/ + l) + 2(z-2)=0, or 23a; -Sy -22=27, 



67.] TANGENT PLANES. 67 

Examples. 

1. Find the tangent plane at (2, — 1, 1) on the surface 
x 2 -2y 2 + z=3. Ans. 4x + 4y+z=5. 

2. Find the tangent plane at ( — 1, 1, 2) on x 2 + y 2 — z 2 — yz—4xy 
=0. Ans. 6x—4y + 5z=0. 

3. The tangent plane at any point of the central quadric 
x Va 2 + y 2 /b 2 + z 2 /c 2 = l is x x x/a 2 + y{y/b 2 + z x z/c 2 = 1 . 

4. The tangent plane to the surface xyz=a 3 makes with the 
coordinate planes a tetrahedron of constant volume. 

5. What are the tangent planes at the origin of the conoid 
x 2 —y 2 =x 2 z 2 ? Ans.x±y=0. 

6. Find the tangent planes to x 4 + y 4 + z 4 = 3axyz, (1) at (0, 0, 0), 
(2) at (a, a, a). 

Ans. (1) x=6, y=0, z=0, (2) x + y + z=3a. 
Also to x 3 + y 3 -{-z 3 =3a 2 £ at the same points. 

Ans. (1) x=0, (2) y + z=2a. 

7. The sum of the squares of the intercepts of the tangent 
planes of the surface x$ + y$ + z =rf on the axes is a 2 . 

8. x =a sin nz, y =a cos nz are the equations of a helix (screw- 
thread) on a circular cylinder of radius a, the 2-axis being the 
axis of the cylinder. Show that the equations of the tangent 
at any point are 

s-*i y-yi 9 9 
= = z —z ly 

ny x —nx x 

and that the tangent makes a constant angle with the £2/-plane. 

9. Find the direction angles of the tangent to the curve of 
intersection of the surfaces of Ex. 6 at the point (a, a, a). 

Ans. 0, 45°, 135°. 



CHAPTER XV. 
SUCCESSIVE DIFFERENTIATION. 

68. Successive derivatives. The differential of fix) is 
(§ 27) /'Or) dx. Let df(x) = f'(x) dx, df'{x) = f"{x) dx, etc. 

The several functions fix), f'(x), f"(x), ... are called 
the first ; second, third, . . . derived functions, derivatives, 
or differential coefficients of f(x). 

Ex. 1. }{x)=3x z -2x + 4, f(x)=9x 2 -2, f'(x)=18x, f'"( x )=18, 

/iv(x)=0. 

2. fix) =sin x, f(x)=cos x, fix) = —sin x, fix) = —cos x, etc. 

3. /(x)-Iog(l+x), f(x)=l/(l + x),f'(x) = -l/(l+xy, etc. 

4. /(a;) =e*, f(x) =e*, f'(x) =e? 3 etc. 

69. Successive differentials. Let y or fix) be a function 
of x, then dy = f(x) dx. The differential of ch/, or d(dy), is 
written d 2 ?/ (read d-two y, or second dy); similarly, 
d s y = d(d 2 y). 

Unless the variable x is given as a function of another 
variable it is assumed to be an independent variable — one 
to which arbitrary values may be assigned; dx is then an 
arbitrary increment of x. It is customary to take dx as of 
the same value in each successive differentiation, i.e., to treat 
the differential of the independent variable as a constant 
in differentiating. Hence, differentiating dy=f(x)dx, 

68 



68-70.] SUCCESSIVE DIFFERENTIATION. 69 

d 2 y=]"(x)dx.dx = f"(x) dx 2 , or d 2 y/dx 2 = f"(x)* (1) 
Hence also, d?y = f"{x) dx 3 , or d s y/dx s = f'"(x), 

and similarly for the higher differentials. Thus the successive 
differential quotients are equal to the successive derivatives. 

Ex.1. y=e ax , dy=ae ax dx, d 2 y=a 2 e ax dx 2 , d 3 y=a 3 e ax dx 3 , .., 
d n y=a n e ax dx n . 

2. y = cos x, dy/dx== —sin x= cos I x+—) , 

.*. d 2 y/dx 2 = cos I x + 2— J , . . . , d n y/dx n = cos (z + n-^j . 

3. y=x n , n a positive integer, d n y/dx n =n\ 

70. If x is not the independent variable — if it is itself a 

function of another variable — we cannot treat dx as a constant 

in successive differentiation. For if x = f(6), dx = f'(0)dd, 

and is therefore a function of d. 

dy 
Differentiating both sides of j r {x) = ~-, we have 

f'(x)dx= dxd2y ~f yd2x , by(D), 

« 

• i"(y\— y—ctya x 

• • / w- dx s • u; 

Comparing with § 69 (1) we see that the d 2 y/dx 2 obtained 
when x is the independent variable is equal to 

dx d 2 y — dy d 2 x 
dx?~ ' 

obtained when x is not the independent variable. 

Ex. The cycloid x=a(d—sin 6), y=a(l — cos 6). Considering 
y as a function f(x) of x, to evaluate /'(x) and f"(x) when 6 =n, ■ 

dx=a(l — cos 0) dd, dy=a sin 6 dd, 

* Since dx is of arbitrary value it may be taken as infinitesimal, 
in which case d 2 y is in general an infinitesimal of the same order as dx 2 . 



70 



INFINITESIMAL CALCULUS. 



[Ch. XV, 



and, being independent variable, 

d 2 x=a sin dd 2 , d 2 y =a cos dd 2 . 



,., x dy sin 6 

:. '(x)=/=- 

dx 1 —cos 



when 6 =n. 



Substituting in (1), f"(x) = — 



1 1 

a{\ —cos 6) 2 4a 



Examples. 

1. y=ax 2 + bx-\-c, dy/dx=2ax + b, d 2 y/dx 2 =2a. 

2. y = (a + x) 3 , dy/dx=3(a + x) 2 , d 2 y/dx 2 =6(a+z). 

3. y =x 2 log x, d 3 y/dx 3 =2x~ 1 . 

4. y = cos ax, d 4 y/dx 4 =a 4 cos ax. 

5. x=sm~ 1 y, d 2 x/dy 2 =y(l —y 2 )~%, d 2 y/dx 2 = —y. 

6. If fix) = sin x, )^(x) =sin (# + w— J . 

7. f(x)=xe?, J (n Hx) = (x + n)e x . 

8. 2/= log re, d n y/dx n = ( — l) n ~ l {n — l)\/x n . 

9. If 2/ = a cos nx + 6 sin nx, d 2 y/dx 2 + n 2 y =0. 

10. If y=ae nx + be- nx , d 2 y/dx 2 -n 2 y=0. 

11. If y=e~ x cosx, d 4 y/dx 4 + 4:y=0. 

12. If 2/=sin -1 £, (l—x 2 )d 2 y/dx 2 —xdy/dx=0. 

13. If 2/=tan -1 :r, (l + £ 2 )d 2 ^/<i:c 2 -f2;c dy/dx=Q. 

14. If 2/=^ sin x, d 2 y/dx 2 — 2dy/dx + 2y=0. 

15. Given x dy—y dx =r 2 dd, show that 

xd 2 y -y d 2 x=2r dr dd + r 2 d 2 0. 

16. By differentiating 

dx = cos <£ ds, and dy = sin 96 #s, 
show that 

(d 2 x) 2 + (d 2 y) 2 = (d<f> ds) 2 + (d 2 s) 2 . 

17. ?/ 2 =4ax, 2y dy =4a dx, or y dy =2a dx; differentiating again, 
y d 2 y + dy 2 =0 (dx being constant), 

.*. y d 2 y+ (2adx/y) 2 =0, or d 2 y/dx 2 =—4:a 2 /y 3 . 

x 2 y 2 

18. Given the ellipse — + - =1, show that 



a 2 ' b 2 



dy 
dx 



b 2 x d 2 y 
a 2 y dx 2 



V 



a 2 y d ' 



70.] SUCCESSIVE DIFFERENTIATION. VI 

d 2 y/dx 2 may be found as in Ex. 17, or from dy/dx. Thus 

dy 
d 2 y b* y X dx 



dx 2 a 2 y 



Substitute for dy/dx and reduce. 

19. If y =f(x) find /"(#)> given x=a cos 0, y=b sin 0, 

Ans. f"(x) = —b 4 /a 2 y\ 

20. a 2 + y 2 =2xy, d 2 y/dx 2 =a 2 /(x—y) 3 . 

21. x 3 + y 3 =3axy, d 2 y/dx 2 =2a 3 xy/(ax — y 2 ) 3 . 



j 



CHAPTER XVI. 



RATES. 




71. Let y be a function of x of which Fig. 44 is the graph. 
When x increases by the amount Ax the change in y is dy, 
and Ay I Ax is called the average rate of change of y per unit 

of x (or briefly, the average 
z-rate of y) fof the change 
Ax in x. When Ax is taken 
smaller and smaller and = 
the average rate Ay/ Ax is 
taken for a gradually dimin- 
ishing change in x, and the 
limit of Ay I 'Ax, namely dy/dx, 
is defined to be the x-rate of 
y for the value x of the vari- 
able. Thus as x increases and reaches the value OA , the 
z-rate of the function y is dy/dx or tan <£>. This is an in- 
stantaneous and variable rate, and is the same as the con- 
stant rate which y would have if P should henceforward 
move along the tangent PD. 

If y=f(x), dy/dx = f ; (x); hence the £-rate of f(x) is /'(#), 
and for a similar reason the x-rate of any derivative is the 
succeeding derivative. 

72. If y is a function of x, as x changes the function will 
increase or decrease according as its graph rises or falls, 
that is, according as dy is + or — . Also dx is + if x in- 
creases. Hence as x increases, y increases or decreases 
according as dy/dx is + or — . Thus a + value of the rate 

72 



71-74.1 RATES. 73 

implies that a function is increasing as its variable increases, 
and a — value implies that the function is decreasing as the 
variable increases. 

73. If x and y are functions of a third variable t, 

dy/dx= (dy/dt)/ (dx/dt). 

Hence dy/dx is the quotient of the simultaneous rates of 
change of y and x, or dy and dx are proportional to the rates 
of y and x.* 

If y = f{x) y and x is a function of t, 

dy/dt = f (x) . dx/dt, 

which gives the rate of y in terms of that of x. ' 

If u = f(x, y), and x and y are functions of t, then, § 45 (5), 

du _dudx du dy 
dt dx dt dy dt ' 

which gives the rate of u in terms of the rates of x and y. 

74. If a point moving in a straight line is at a distance x 
from a fixed point in the line at the end of an interval of 
time whose measure is t, its velocity v is the £-rate of x, and 
is therefore dx/dt; and its acceleration a is the £-rate of v, 
and is therefore dv/dt. But 

1 (dx\ 
dv_ \dt I _d 2 x ., dv__dvdx_dv 
dt dt dt 2 ' ' dt dx dt dx 

■ dx , dv d 2 x dv 

Hence v=-r: and a = — =—r^ = v -r~- 

at dt dt z dx 

Similarly the angular velocity and angular acceleration of 

a revolving body are -7- and -7-^ respectively. 

Time rates are sometimes indicated by dots, x being the 
same as dx/dt, and x the same as d 2 x/dt 2 . 

* In some treatises dy and dx are defined to be rates. 



74 INFINITESIMAL CALCULUS, [Ch. XVI. 



Examples. 



1. The ordinate of the curve y = V25—x 2 is moving parallel 

to the ?/-axis at the rate 2 in. per sec. At what rate is its length 

changing when x = 3 ? 

dy x dx 3 _ . _ . n ^ T 

-7- = .. = — — when x =3 and dx /at = 2. Hence y is 

decreasing at the rate of \\ in. per sec. 

2. At what points on the curve y= log sec # do £ and y change 
at the same rate? Arts. x = (n + \)7i, n an integer. 

3. Find the acceleration if (1) v=u + bt, (2) x=ut + bt 2 , (3) 
v 2 =u 2 + bx, u and b being constants. Arts. (1)6, (2)26, (3)^6. 

4. If z=a cos (&£ + c), show that the acceleration = — b 2 x. 

5. If x=a sinh (bt + c), show that the acceleration = b 2 x. 

6. Show that tan x always increases with x. 

7. Three adjacent sides of a rectangular parallelepiped are 
3, 4, 5 inches in length, and are each increasing at the rate of 
'02 in. per in. per min. At what rate is the volume increasing? 

Ans. 3*60 cu. in. per min. 

8. One end of a ladder moves down a vertical wall with velocity 
v 1} while the other end moves along a horizontal plane with veloc- 
ity v 2 . Show that v 1 /v 2 =tan 0, where 6 is the angle which the 
ladder makes with the vertical. 

9. Two straight lines of railway intersect at an angle 60°. On 
one a train is 8 miles from the junction and moving towards it 
at the rate of 40 miles per hour, on the other a train is 12 miles 
from the junction and moving from it at the rate of 10 miles 
per hour. Is the distance of the trains from each other increasing 
or decreasing ? 



CHAPTER XVII. 



MAXIMA AND MINIMA. 



75* Suppose y to be a function of x and that x continually 
increases. Then (§ 72) y will increase or decrease accord- 
ing as dy/dx is + or — . When dy/dx changes from + to 
— , y ceases to increase and begins to decrease, and is then 
said to be a maximum; when dy/dx changes from — to +, 
y ceases to decrease and begins to increase, and is then 
said to be a minimum. Now in order that a quantity may 
change sign it must become or 00 or — go ; * hence as y 
becomes a max. or a min., dy/dx becomes or 00 or —00 
and changes sign from + to — for a max. and from — to + 
for a min. 




Fig. 45. 

76. Suppose, for example, that the curve of Fig. 45 repre- 
sents the graph of a function and that it is traced by a point 
moving from left to right so that dx is + . Then y decreases 
from A to B and dy/dx is — , between B and C y continually 

* A quantity may change sign on account of finite discontinuity 
without passing through the value 0, but this occurs so rarely that 
we need not consider it further. 

75 



76 INFINITESIMAL CALCULUS. [Ch, XVII. 

increases and dy/dx * is + ; at B y ceases to decrease and 
begins to increase, dy/dx changes from — to + through the 
value 0, and y is a min. Similarly at C y is a max., and again 
a min. at D. At E dy/dx becomes oo and changes from + 
to — , hence y is a max., and similarly y is a min. at F. 

Points such as A, B, etc., are called turning points) and 
the max. and min. values of y are called turning values. 

It will be noticed that a max. is not necessarily the greatest 
of all the values of y; it is greater than the values which im- 
mediately precede or follow it; and similarly a min. is not 
necessarily the least value of y. 

77. To obtain the values of x which make a function y a 
max. or min. we must obtain dy/dx and find what values of 
x cause it to become zero or infinite. To distinguish the 
maxima from the minima we must determine whether dy/dx 
changes from + to — or from — to + as x passes through 
the critical value. In the former case y will be max., in 
the latter a min. It may happen, however, that dy/dx does 
not change sign, although it becomes or 00 (e.g., at G, 
Fig. 45), in which case y is neither a max. nor a min. 

Ex. 1. y=x*-6x 2 + 9x + l. 

Here dy/dx =3x 2 -12x + 9=3(x-l)(x -3). 

When x is a little less than 1, x — l is — and x—3 is — , 
,\ dy/dx is +. 

When x = l, dy/dx is 0. When x is a little more than 1, re — 1 
is + and x—3 is — , .'■ dy/dx is — . Hence dy/dx changes from 
+ to — through and .'. y is a max. when x = l. Substituting 
1 for x in the given function we find the max. value of y to be 5. 
Similarly x =3 makes y a min., viz., 1. 

2. y = (x — l) 3 , dy/dx=3(x — l) 2 ; .*. dy/dx =0 when x = l, but 
does not change sign f when x passes through this value, /. y 
is neither a max. nor a min. 

* It will be remembered that dy/dx = tsm <j>, and is therefore -f- 
or — according as <£ is + or — . 

f (x — a) n changes sign with x—a only when n is an odd integer, 
or a fraction whose numerator and denominator are both odd 



77-79.] MAXIMA AND MINIMA. 77 

2 

3. y*=2 + (x — l)a, dy/dx = -r- —7; .". dy/dx becomes —00 

o \X 1 J 3 

and changes from — to -f as x passes through the value 1, hence 
x = l makes y a min., viz., 2. 

78. The sign of d 2 y/dx 2 ( = the z-rate of dy/dx) tells us 
at any time whether dy/dx is increasing or decreasing. If 
then the value of x which makes dy/dx equal to also makes 
d 2 y/dx 2 plus, we infer that dy/dx is increasing when it passes 
through 0, i.e., that dy/dx changes from — to +, and hence 
that y is a min.; whereas, if the value of x which makes 
dy/dx equal to also makes d 2 y/dx 2 minus, we infer that 
dy/dx is decreasing when it passes through 0, i.e., that it 
changes from + to — , and hence that y is a max. 

Hence to distinguish the maxima from the minima we may 
find d 2 y/dx 2 , and in it substitute the values of x which make 
dy/dx equal to 0. Then for every -f- result y is a min., and 
for every — result y is a max.* 

Ex. 1. In Ex. 1, § 77, d 2 y/dx 2 =6x — 12, which is — when x = l, 
and + when x=S. Hence x = \ makes y a max. and x=3 makes 
y a min. 

2. y=x 3 —7x 2 + 8x + 30, dy/dx =3x 2 — 14z + 8. For a max. or a 
min. 3x 2 — lix + 8=0, .'. x=\ or 4. 

Also d 2 y/dx 2 = Qx — 14, which is — when £=§ and + when 
x =4; .*. x =f makes y a max. and x =4 makes y a min. 

79. It should be noticed: 

(1) That max. and min. values must occur alternately in 
a continuous function, i.e., between two successive max. 
values there must be a min., and between two successive 
min. values there must be a max. Also of two values of x 
which make y a max. or a min., if one makes it a max. the 
other must make it a min. 

(2) When y has a turning value, y n (n a positive or nega- 
tive integer) has a turning value. Thus a square-root sign 

* If d 2 y/dx 2 is or 00 it gives no information as to the turning 
values, and the test of § 77 must be applied. 



78 



INFINITESIMAL CALCULUS. 



[Ch. XVII. 



affecting the whole of the variable part of a function may 
be disregarded in differentiating. 

(3) A constant factor may be omitted from the function 
before differentiating, since it cannot affect the values of x 
for which the derivative is or oc . 



Ex. y = 7r£V / a 2 — x 2 . This =nVa 2 x 2 — x 4 , and .'. y will be a 
max. or a min. when a 2 x 2 — x 4 is a max. or a min.; hence 2a 2 x 

-4a; 3 = 0, .'. s = 0, and x= ±a/V2. 

8o. In the practical applications of this subject it will be 
necessary to form the function which is to have a turning 
value. It will frequently be obvious from the nature of the 
problem whether the result corresponds to a max. or a 
min. 

Ex. 1. Of all arithmetical fractions, which one exceeds its square 
by the greatest quantity? 

Let the fraction be x. Then x — x 2 is to be a max. 

.*. 1— 2x = 0, and hence x = J. 

2. How to make with a given amount (area) of material a 
cylindrical box (with lid) which shall have the greatest possible 
volume. 

We have the total surface of the cylinder given, call it s, and 
assume h for the height and x for the radius of the base. 

Then s = 2nx 2 + 27ixh, .\ h = s/(2nx)—x. . 

The volume V = 7ix 2 h = ^sx — nx 3 . 

.". dV /dx = \s — 37nr 2 = t for a max. 

.*. x = \ / s/67i ) whence h = 2\ / s/67z. 

Hence the height must = the diameter of the base and each 



= 2Vs/6tt. 

[Observe that in these examples the function which is to be a 
max. or a min. must be expressed in terms of some one variable 
with or without constants; in this case the function is 7:x 2 h, 
where both x and h are variable, but there is a relation connect- 
ing x and h from which h may be obtained in terms of x; this 
when substituted in nx 2 h gives a function with one variable.] 



SO] 



MAXIMA AND MINIMA. 



79 



3. To find the greatest isosceles triangle that can be inscribed 
in a given circle. 

Let ABC (Fig. 46) be an isosceles triangle inscribed in a circle 
of radius a and centre E. Let DC = x. Then 

AD = VaE^-DE 2 = Va 2 ~(x-a) 2 = V2ax-x 2 . 
.'. area of ABC=LC . AD = xV2ax-x 2 = V'Zax i -x\ 

This will be a max. when 2ax 3 — x { is a max., §79(2), i.e., 
when 6ax 2 — 4:X 3 = 0, .*. x = §a. 

The triangle is easily shown to be equilateral. 

4. One corner A of a rectangular piece of paper ABCD (Fig. 47) 
is folded over to the side BC. Find when the crease EG is a min. 

Let AB = a, AE = x, EG = y, AGE = 6. Then BEF = 2d. 
.'. BE:EF=(a-x)/x=cos20 and AE\ FG=x/y = sm 6. 
Eliminating d by the relation cos 26 = 1—2 sin 2 0, we find 

y 2 = 2x 3 /(2x-a), 

from which y is found to be a min. when x = \a. 






Fig. 46. 



Similarly it may be shown that the area of the part folded over 
is a min. when x = fa. 

5. To cut the parabola of greatest area from a given right 
circular cone, Fig. 48. 

Let AB = a and FB = x. The area = f ED . FG. 

Now EF 2 = AE . EB= (a-x)x, and ED is prop ortional to x. 

.\ area varies as xV( a -x)x or Vax 3 -x 4 , whence x = \a 
for a max 



~.--- 



80 INFINITESIMAL CALCULUS. [Ch. XVII. 



Examples. 

1. x 2 — 3x + 4:, min. when £ = §. 

2. x 5 — 5x 4 -\- 5x 3 -\- 1 , max. when x = \ y min. when x = 3. 

3. a + 6(c— oO^, no turning value. 

4. x 3 — 2x 2 — 4:X + l, max., when #= — §, min. when £ = 2. 

5. (x — l) 3 (x + 2) 4 , max. when #= —2, min. when z = — f. 

6. (1 +3z)/\/4 + 5£ 2 , max. when x = ± 5 2 -. 

7. (:c + 2) 3 /(a; — 3) 2 , min. when £ = 13. 

8. sin d + cos (9, max. when &=\n, min. when Q = \n. 

9. sin 0/(1 +tan 0), max. when = \7i, min. when Q = ^n. 

10. sin # sin(a — d), max. when # = -|«. 

11. sin 2 # cos 3 #, max. when sin 6= ±Vf, min. when = 0. 

12. Min. value of a tan d + b cot 6 = 2\/~ab. 

13. Min. value of a 2 sec 2 # + & 2 cosec 2 # = (a + 6) 2 . 

14. Min. value of ae nx + be~ nX = 2\ r ab. 

15. Max. value of log x/x = l/e. 

16. What is the longest ordinate of the curve a 2 y 2 = x 2 (a 2 — x 2 ), 
(Fig. 69)? Arts. £a. ' 

17. Find the max. ordinates of the curves 

(y-x) 2 = x 3 , Fig. 30, and (y-x 2 ) 2 = x\ Fig. 31. 

Ans. 2 2 /3 3 , 4 4 /5 5 . 

18. Find the max. ordinate of the curve 

x 3 + y 3 = 3axy, Fig. 28. 

Differentiating the equation and making dy = we have x 2 = ay; 
from this and the equation of the curve we find the max. ordinate 
to be at the point (a ^/2, a ^4), the latter coordinate being the 
required value. 

19. Find the max. ordinate of the curve y 3 = x 5 + Sax 2 , Fig. 38. 

Arts. i/~ia. 

20. How could you cut out four equal squares from the corners 
of a given square so that the remaining area (the edges being 
turned up) would form a rectangular box of greatest volume? 

Arts. Each side of the little squares = \ of a side of the given 
square. 

21. Find the breadth and depth of the strongest beam that can 
be cut from a cylindrical log of diameter d, assuming that the 



80.] MAXIMA AND MINIMA. 81 

strength varies as the product of the breadth and the square of 
the depth. 

Ans. Breadth = i^3 d, depth = J^6 d. 

22. To cut out from a given sphere the cone of greatest volume. 

Ans. Ht. of cone = f diam. of sphere. 

23. How could you cut a sector out of a circle so that the re- 
mainder of the circle would form the lateral surface of a cone of 
max. volume? Ans. Leave Vf of circumference. 

24. What is the shortest distance of the line y = x + 2 from the 
parabola y 2 = 4:X? Ans. ^V2. 

25. Assuming that the work of propelling a vessel in still water 
varies as the cube of the speed, what is the most economical rate 
of steaming against a current of speed v ? 

The expense for a given distance varies as x 3 and the time, and 
the latter varies inversely as x — v. Ans, \v. 



CHAPTER XVIII. 



CURVATURE. 



8 1. Direction of curvature. Let it be supposed that the 
tangent of a curve rolls round the curve in such a way that 
the abscissa of the point of contact P continually increases. 
Let the tangent make an angle with the x-axis. Then a 
+ value of d 2 y/dx 2 (the x-rate of dy/dx) at P implies that 
dy/dx or tan <£, and therefore also (f>, is increasing with x, 
or that the tangent is turning in the positive direction as x 
increases. In other words, the curve bends upward, or is 
concave upward, when d 2 y/dx 2 is + , and bends downward, 
or is concave downward, when d 2 y/dx 2 is — . 

82. Point of inflexion. A point where a curve has ceased 

to bend upward and is about 
to bend downward, or vice 
versa, is called a point of in- 
flexion. At such a point d 2 y/dx 2 
must change sign, and must 
therefore become 0, 00 , or — 00 .* 

FlG - 49, A tangent at a point of in- 

flexion is sometimes called a stationary tangent, for, if the 




* It is assumed in the above that x is the independent variable. 
If y is the independent variable, d 2 x/dy 2 must change sign. If neither 
x nor y is independent, the quantity which must change sign is (§70) 



(dx d 2 y—dy d 2 x)/dx 3 . 



82 



81-83.] 



CURVATURE. 



83 



tangent is supposed to roll round the curve, it comes to rest 
at such a point and reverses its motion. 

83. If P, Fig. 50, is a point of inflexion, the secant through 
P and a point Q near P also passes through another point 
Q' near P. As the secant approaches the position of the 
tangent at P, Q and Q' approach coincidence with P at 
the same time. Hence the inflexional 
tangent is sometimes said to pass 
through three coincident points of 
the curve. A tangent at an ordinary 
point on a curve of the nth degree 
cannot meet the curve in more than 
n — 2 other points; the tangent at a 
point of inflexion cannot meet the 
curve in more than n — 3 other points, 
and in not more than n — 4 other points if the point of con- 
tact is also a double point (as in Fig. 27). 




o 



Fig. 50. 



Ex. 1. y=(x-l) s , d 2 y/dx 2 = 6(z-l). This is - when x<\, 
when x = l, + when x>l; hence, as x increases, the curve bends 
downward until x=l, and upward afterwards; .*. there is a 
point of inflexion where x = l. Since y and dy/dx are also 




Fig. 51. 



Fig. 52. 



Fig. 53. 



when £=1, the axis of x is the tangent at the point of inflexion 
(Fig. 51). 

2. y={x-\)\ d 2 y/dx 2 = 12(x-l) 2 , which is when x = l, 
but is never — hence there is no point of inflexion (Fig. 52). 



j ' 



84 



INFINITESIMAL CALCULUS. 



[Ch. XVIII. 




Arts. (0, 0). 
(-1,0). 
(2a, fa). 

(1, 0). 

(a, 0). 



3. y 3 = x, or y = x$, d 2 y/dx 2 = — faHl, which becomes oo and 
changes from + to — when x = 0, .'. the origin is a point of 
inflexion (Fig. 53). 

4. y = 3x*-4x*-6x\ d 2 y / dx 2 = 12 (Sx 2 -2x-l). Putting this 
= and solving for x } we get x= —^, x = l, which determine the 
points of inflexion. 

Find the points of inflexion on the curves: 

5. a 2 y = x(x 2 -a 2 ), Fig. 17. 

6. xy= 1 +x 3 . 

7. (x + a) 2 y = a 2 x. 

8. y = x(x-l)(x-2), Fig. 70. 

9. x* — axy = a 3 . 

10. (a 2 + x 2 )y = a 2 x. 

(0, 0), (±aV3, ±Ja\/3). 

11. ?/ 3 = x 3 + 3a:r 2 , Fig. 38. (-3a, 0). 

12. z 3 + ?/ 3 = a 3 . (a, 0), (0, a). 

13. x = y 3 + 3y 2 . (2, -1). 

14. 2/ 2 = z 2 (2;c-l), Fig. 33. (f, ±|V3). 

15. Show that at a point (x, y) a curve is con- 
vex or concave to the axis of # (i.e., with reference 
to the foot of the ordinate) according as y d 2 y/dx 2 
is + or — . 

16. Show that the curves y = sinx, y = tsaix f 
meet the axis of x in points of inflexion. 

17. Where are the points of inflexion of the 
curve y = cos a;+| cos Sx ? 

Ans. Where x = \%n ) n any integer not divisible 
by 4. 

18. On the witch y 2 (a — x) = a 2 x (Fig. 54), show that the points 
of inflexion are (a/4, ±a/v3). 



Fig. 54. 



84. Centre, radius, and circle of curvature. Let P and 

Q be two points near one another on a curve APQE, 
Fig. 55, at which tangents and normals are drawn, the 
latter meeting in D. The limit of position C which D 
approaches as Q moves towards coincidence with P is 
called the centre of curvature of the curve at P, PC is 



84-86.] 



CURVATURE. 



85 




Fig. 55. 



called the radius of curvature, and the circle with C as 
centre and PC as radius is 
called the circle of curva- 
ture. 

The extremities of an infini- 
tesimal arc are called consec- 
utive points of the curve.* 
The normals at consecutive 
points are consecutive nor- 
mals. Hence the centre of 
curvature is the limit of the 
point of intersection of con- 
secutive normals. 

85. Let the length of PC 
be R. Let the tangents at P, 

Q make angles </>, <fi + J(j) with OX, then d<fi = PDQ. Let 
s = the length of the arc of the curve measured from some 
point up to P, Js = the arc PQ, and g = the chord PQ. Then 
PD/q=smPQD/smJ(j). The limit of sinPQD=l, since 
the limit of PQD is a right angle. Hence the limit of PD = 

£(q/smJ^) = £(Js/Jcf>) (§17) =ds/dcf>. 
.*. R=ds/d(f>. 

86. Imagine the tangent to be rolling round the curve, 
the point of contact having arrived at P. Then dcf>/ds is 
the s-rate of <£, or the rate, in radians per unit length of 
the curve, at which the tangent is turning. This rate is 
taken as the measure of the curvature of the curve; hence 
1/R measures the curvature at P. Since all normals of a 
circle intersect in the centre and are equal to the radius, 
the curvature of the circle of curvature is constant and 
= 1/P. 



* The point consecutive to P is the point which is next considered 
and supposed subsequently to approach coincidence with P. 



86 INFINITESIMAL CALCULUS. [Ch. XVIII. 

87. The circle of curvature generally crosses the curve at 
the point of contact, since in the circle the curvature is the 
same on both sides of the point of contact, which is not the 
case in the other curve except possibly at certain points, 
e.g., at the vertex of a conic section, where the circle of curva- 
ture does not cross the curve. 

88. Length of the radius of curvature. We have seen that 

R = — (1); we also have -^ = tan<£. (2) 

Differentiating (2), 



2 



dx d 2 y — dy d 2 x , , , fds\ 
—^ -BecV #= [jj d4>, 

j. dx d 2 y — dy d 2 x 
•"• «9= —j^t > ( 3 ) 

ds s 
dx d 2 y — dy d 2 x 

We may generally take x as the independent variable 
and therefore make d 2 x = 0; also ds 2 = dx 2 + dy 2 . 

R _ (dx 2 + dy 2 f _ l 1+ \dx) J 
dx d 2 y * d 2 y 

dx 2 

The sign of R when found from (5) will be + or — accord- 
ing as d 2 y/dx 2 is + or — , that is, according as the curve is 
concave upward or concave downward (§ 81). 

If x and y are given in terms of a third variable m which 
is taken as independent, (4) may be expressed in the form 



rfe% 



/dy\ 2 -|f 
_ L ^dml ' \dmJ J ( „. 

dx d 2 y dy d 2 x 



dm dm 2 dm dm 2 



87, 88.] 



CURVATURE. 



87 



Ex. 1. To find the radius of curvature at any point (x, y) of 

the ellipse — , + \, = 1, Fig. 56. 
a 2 b 2 

By differentiating the equation 

of the ellipse we have 

b 2 x d 2 y b* 

a 2 y dx 2 



dy 
dx 



a 2 y 3 



Substituting in (5), we have 



R = 



4M 



a 4 6 



which gives R in terms of x and 
?/.* A more convenient expres- 
sion may be found by substi- 
tuting y 2 from the equation of 
the curve. 




Fig. 56. 



Then 



R = 



(a 2 — e 2 x 2 ) 
ab 



where e is the eccentricity \/a 2 — b 2 /a. 

It is known that (a 2 — e 2 x 2 )^ = the semi-diameter parallel to the 
tangent, or perpendicular to the normal, at (x, y). 

Calling this b x we have 



R=- 



ab' 



(7) 



2. To find R at the origin of the curve ay 3 — 3ax 2 y = x i , Fig. 36, 
for the branch which touches the x-axis. 

Let y = mx,\ then x = a(m 3 —3m), y = a(m i — 3m 2 ). 

Thus x and y are known in terms of a third variable, and we 
require R from (6) for m = 0. Differentiating, 

dx/dm = a(3m 2 — 3)= —3a for ra = 0. 
d 2 x/dm 2 = 6am = for ra = 0. 

dy/dm=a(4:m 3 — 6m) = for m = 0. 
d 2 y/dm 2 = —6a. 

■* The sign of R will be + or — according as y is -J- or — . 
f See foot note, p. 53. 



88 



INFINITESIMAL CALCULUS. 



[Ch. XV II. 



Whence, from (6), R = %a. 

Similarly ft = 24a for the other branches of this curve at the 
origin (ra = V3). 

89. Coordinates of the centre of curvature. Let the co- 
ordinates be a and /?. Then, Fig. 57, 




dx d 2 y — dy d 2 x 



a = x — R sin cf> 

J? dy_ dy ds 2 

— X xL ~ — X 

as 
{3 = y + R cos <f) 

ds dx d 2 y — dy d 2 x 



• (1) 



. (2) 



Fig. 57. 



Evolute — Involute. The locus of 
the centres of curvature of a curve 
is another curve which is called the evolute of the given 
curve, and the given curve is called the involute of the 
evolute. 

Ex. To find the centre of curvature for any point (x, y) of an 
ellipse, and the equation of the evolute. 

If x is the independent variable, (1) and (2) become 



dy \dx 
dx d 2 y 
dx 2 



1 + 



dy 
dx, 



<Py ' 
dx 2 



which for the ellipse give 



a 2 -b' 



y- 



Solving for x and y and substituting in the equation of the ellipse 
we obtain 

(aa)i + (bp)$=(a 2 -b 2 )i 

for the equation of the evolute (see Fig. 56). If x and y are sub- 
stituted for a and /?, the equation becomes 

(ax)i + (by)i=(a 2 -b 2 )*. 



89, 90.] 



CURVATURE. 



89 



90. Properties of the evolute. (1) Every normal of a 
curve touches the evolute at the centre of curvature. 



a = x — Rs'mcj), .'. da = dx — R cos cf> d<f> — sin cf> dR. 
But dx=ds cos (f>=R dcf> cos <j>. 

.'. da= —sin <p dR. 
Similarly d/?= cos $ dR. 

.'. dp /da = — cot (f) = — dx/dy. 



(i) 

(2) 



Hence the tangent of the evolute at {a, /?) has the same 
slope as the normal at (x, y) on the involute, and (a, /?) 
is on both lines, therefore they coincide. 




Fig. 58. 

(2) As long as the radii of curvature of a curve continue 
to increase or to decrease, the difference of any two is equal 
to the arc of the evolute included between them. 

Let the arc of the evolute be S. Then 

dS=Vda 2 + d(3 2 =±dR, 

from (1) and (2). Suppose R to be increasing. Then dS = dR, 
hence aS and R can only differ by a constant, and therefore 
any increment of S is equal to the corresponding increment 
of R. Similarly if R is decreasing, the increment of S is 
equal to the decrement of R. 



90 



INFINITESIMAL CALCULUS. 



[Ch. XVIII 



From these properties of the evolute it will be obvious 
that if one of the tangents of the evolute were supposed to 
roll round the curve, a tracing-point in it would describe 
the involute. Thus although a given curve can have only 
one evolute, it can have any number of involutes. The 
involute might also be described by a tracing-point in a 
string which is kept stretched at the same time that it is 
unwound (" evolved ") from the evolute. 



Ex. The radii of curvature at the extremities of the axes of an 
ellipse are, § 88 (7), a 2 /b and b 2 /a. Hence the whole length of the 
evolute is 

V_6^\ /a 3 -6 3 

b a ] \ ab 



4 =-- 



')• 



Examples. 



1. At any point of the parabola y 2 — 4ax show that R = —2Vr 3 /a, 
where r is the focal distance ( = a + x) of the point. Hence it may 




Fig. 59. 

be shown ths*,t R = twice the intercept on the normal between the 
directrix and the curve. 

2. Prove that for y 2 = Aax, a = 2a + 3x, /9= -y 3 /4:a 2 . 



90.] 



CURVATURE. 



91 



3. Show that the e volute of the parabola is the semi-cubical pa- 
rabola 27 ay 2 = 40- 2a), Fig. 59. 

4. Show that C, C are (8a, ±4v / 2a), and that they are the 
centres of curvature of B' f B. 

5. Show that the arc AC = 2a(3v / 3-l). 

6. Show that R = (1 -fa 2 )§/26 at the origin on the curve 



or 



y = ax + bx 2 + ex 3 4- . . . 
x = ay + by 2 + cy 3 + . . . 

7. Find R at the origin of the following curves: 

(1) The parabola y 2 = 4cax, or x 2 = 4:ay. 

(2) y 2 = x 2 (l +2x), Fig. 32. We have 

y = x(l + 2x)$ = x(l +x — ix 2 + . . .). 

(3) y 2 = x\l+2x), Fig. 34. 

(4) (y-x 2 ) 2 = x\ Fig. 31. 

(5) (y-x) 2 = x 3 , Fig. 30. 

8. Find the R of x = y 2 + y 3 when x = 2. Arts, 

9. Find R at the point of maximum ordinate on the curve 
y3 = x 2 + 3ax 2 , Fig. 38. Arts. -fy~2a. 

10. Show that R = 2 \/2a on the branch of the curve ay* —ax 2 y 2 =* 
x b , Fig. 37, which touches y=.x at the origin. 

i 



Arts, 


2a. 


± 


V2 




±h 




h 




0. 


_13\ 


/26. 




Fig. 60 



/A* /v\* 

11. The equation ( -] +(t) =1 represents a common parab- 
ola, the origin being a point on the directrix, and the axes tan- 
gents to the curve. Show that R = 2(ax-{-by)i/ab. 
[Fractional indices may be avoided by using x = a cos 4 #, y = b sin 4 d.] 



92 INFINITESIMAL CALCULUS. [Ch. XVIII. 

12. Find R at any point of the curve x = a ccs } y = c s?^ d. 



13. In the hypocycloid x% + y$ = a$ (Fig. 18) show that R = 3^/axy 
= 3 times the perpendicular from the origin on the tangent (Ex. 
3, p. 46). 

14. Show that the radius of curvature at any point, of the cycloid 
x = a(0 — sin 6), y = a(l—cosd) is —4a sin \Q = twice the normal 
PB (Fig. 60). 

15. Also that a = a(# + sin 0), fl= — a(l — cos 0), and hence (see 
Fig. 20) that the evolute is an equal cycloid. 

16. Show that R = oo at a point of inflexion. 

ds 2 

17. At any point of a curve R = . -_ (See 

V (d 2 x) 2 + (d 2 y) 2 - (d 2 s) 2 

Ex. 16, p. 70.) 



CHAPTER XIX. 
INTEGRATION. ELEMENTARY ILLUSTRATIONS. 

91. Prop. The limit, when n is infinite, of the sum of n 
infinitesimals of the same sign is not changed if the infini- 
tesimals are replaced by equivalent ones (§ 15). 

Let the given infinitesimals be ct\, (X2, • . . and let 
Pi, P2, • • • be equivalents. By a theorem of algebra 
(/?i+/?2 + - • • )/(0L\+a2 + . . . ) lies in value between the 
greatest and the least of the fractions* Pi/cci, P2/&2, ••• 
But each of these fractions = 1 by hypothesis. Hence 

£(/?i+/?2 + . • = £(ai+a 2 + . . . ), 

or £ip=£Za. 

Hence (§ 16) the limit of the sum depends only upon the 
infinitesimals of the lowest order. 

92. In particular, if y is a function, and Ay and dy the 
infinitesimal increment and differential corresponding to 
the infinitesimal increment of the variable, then (§ 42) 
dy = dy + I, and hence £2 Ay = £Z dy. This will be further 
considered in the following article, and a special notation 
will be employed for the limit of a sum. 

* Let fin/an be the greatest of the fractions, and let it = r. Also 
suppose the a's to be all positive. Then 

(3 l <ra l , p><ra 2 , . . . , p n = ra n , . . . 

Hence, the symbol I indicating the sum of all terms of a single type, 

ZP<rZa, or I3/Ia<r. 

Similarly I^/Ia> the least of the fractions. 

93 



94 INFINITESIMAL CALCULUS. [Ch. XIX. 

93. Let F(x) be a function of x, fix) its derivative, and 
suppose F(x) and f(x) to be continuous from x = a to x = b. 
When x changes from a to b the change in F(x) is F(b) — F(a), 
or, in symbols, 

[F(x)J = F(b)-F(a). 

Suppose that x changes by the successive addition of 
infinitesimali ncrements. When x has the increment dx] 
the corresponding increment of Fix) is, § 42 (2), fix) dx + I, 
where / stands for the higher infinitesimals. Hence F(b) — 
i<\a) = the limit of the sum of all such terms as fix) dx, while 
x changes from a to b, dx approaching its limit and the 
number of terms being infinite. Let this sum-limit be ex- 
pressed by fix) dx. Then 

J a 

\ b f(x) dx=[F(x)~f = F(b)-F(a). 

J a L -I a 

Hence, f(x) being a function of x which is continuous from 
x=a to x=b, to find the limit of the sum of all such terms 
as f(x) dx when x changes from a to b we must seek the func- 
tion F{x) of which the differential is f(x) dx, substitute therein 
b and a successively for x, and subtract the second result 
from the first. 

This process, which is analogous to summation,* is called 
integration (the making of a whole from infinitesimal parts) ; 
F(x) is called the integral of fix) dx, and fix) dx is called 
an element of the integral; a and b are called the limits f 

. f ft 

of the integration; f{x) dx is read "integral from a to b 

J a 

(or between a and b) of fix) dx" 

* Historically, the symbol / is the old form of the letter s f the initial 

letter of the word sum. 

t This meaning of the word limit is not the same as that employed 
elsewhere. It here signifies a value of the variable at one end of its 
range. 



93, 94.] 



INTEGRATION. 



95 



It should be noticed that dx is here regarded as an in- 
finitesimal increment of x, and that the element or differ- 
ential f(x) dx is (§ 42) the increment of the function F(x) 




b x 




Fig. 62. 



when the higher infinitesimals are omitted or disregarded. 
The practical applications of integration depend upon the 
fact that the element can be written down when F(x) is 
unknown. 

Illustrations. 

94. Areas of curves. Let y = f(x) be the equation of a 
continuous curve CD. Let OA = a, OB = b, OM=x, MP=y, 
MN=dx, RQ = dy, and let it be required to find the area 
ABDC. 

When x has the increment dx, the increment of the area is 
ATZVQP = rectangle MR + PRQ. MR = y dx, and PRQ<SR 
which = dx Ay and is therefore a higher infinitesimal. Hence 
the element of the area is y dx* 



.'. the area ABDC= 



y dx 



f(x)dx=[F(x)T, 



where F(x) is the function of which the differential is f{x) dx; 

e.g., the function of which the differential is x n dx is ~x n + l 

n + 1 

except when n— — 1, in which case it is log x. 

* Observe that the omissio N ( f the higher infinitesimals is equiva- 
lent to supposing y to remain constant while x increases by dx. More 
generally, the element of dx(P l + \) (P 2 + i 2 ) ... is dx . P 1 P 2 . . . , and 
P,, P 2 . . . may, in obtaining the element, be regarded as constant 
while x changes to x-\-dx. 



96 



INFINITESIMAL CALCULUS. 



[Ch. XIX. 



In other words, we imagine the area to be divided into 
narrow strips by lines drawn parallel to OY ', express the 
area of a strip as a differential or infinitesimal element (all 
infinitesimals of an order higher than the first being omitted) 
and then integrate. The whole area is seen to be the limit 
of the sum of the rectangles as their breadth =0 and their 
number becomes infinite. 

Ex. 1. To find the area OBD of the curve y = x 3 , Fig. 63, the 
limits of x being and 1. 



The area= 



f 1 r i 1 

ydx= x 3 dx=\ \x* =f, i.e., the area is one- 



fourth of the square on OB. 





Fig. 64. 



Fig. 65. 



2. The area of the parabola y 2 = kax, Fig. 64; from x = to 
x = h is 

y dx = 



V4a ,x*dx= f" V4a . fxi~| 

o L 



= \\/±a . B = $h .Vlah = %OB . BD 

= two-thirds of the rectangle having the same base and height. 
3. The area of the curve y = sin x, Fig. 65, from x = to x = n is 



i: 



sin x dx = — cos x = 2, 

o 

i.e., twice the square on the maximum ordinate. 



95. Volumes of solids of revolution. Suppose the curve, 
Fig. 61, to revolve about OX and generate a solid. The 
rectangles MR, MQ generate cylinders of infinitesimal thick- 
ness dx, and radii y, y + dy, and therefore of volume 7zy 2 dx, 



95.] ELEMENTARY ILLUSTRATIONS. 97 

Ti{y + Ay) 2 dx. The latter = ny 2 dx when the higher infinitesi- 
mals are omitted. Hence the volume element = Try 2 dx. 



/. the whole volume 



= 7z\ y 2 dx. 



In other words, we imagine the solid to be divided into 
thin slices by planes perpendicular to OY ', express the volume 
of a slice as a differential or infinitesimal element, and then 
integrate. The whole volume is thus the limit of the sum 
of cylinders of volume ny 2 dx, i.e., of the cylinders formed 
by the revolution of the rectangles of Fig. 62. 

Ex. 1. The volume formed by the revolution of OBD, Fig. 63, 
round OX is 



•f 

J 



y 2 dx = ~ 



1 r T 1 

x G dx = 7t\ lx 7 = ]n. 

o L o 



2. When the area of the parabola y 2 = 4:ax from x = to x = h 
revolves about OX the volume is 



rh 

y 2 dx = n 

o 



h r- -Jl 



±axdx = 7z[Aa.%x 2 I =%n(±ah)h = %7zBD 2 .OB, 
L o 

i.e., one-half of the cylinder having the same base and height. 



CHAPTER XX. 



FUNDAMENTAL INTEGRALS. I. 



96. We have seen that 

'f(x)dx=[F(x)\, 

F(x) being the function of which f(x) dx is the differential. 
If the limits are not expressed,* we may write 

[f(x)dx = F(x), 

and hence may be regarded as a symbol which indicates 

the operation of going from the differential f(x) dx back to 
the primitive function F(x), or of finding the antidifferential 
of j(x) dx, or the antiderivative of f(x). By this operation 

we can discover only the variable part of the primitive 

• 

function; e.g., 2xdx = x 2 , or x 2 + l, or x 2 + c, where c may 

be any constant (any quantity independent of x). To 
every integral thus obtained from a differential there should 
.*. be added a constant, the value of which must depend upon 
special data; we should then write 

f(x) dx = F(x)+c. 



* The integral is said to be definite when the limits are expressed, 
indefinite when they are not expressed. 

98 



96, 97.] 



FUNDAMENTAL INTEGRALS. 



99 



If we are hereafter to substitute limits, the constant c 
need not be .expressed, inasmuch as it would disappear in 
subtracting. We shall accordingly, as a rule, omit the 
constant, but its presence is always understood. 

97. The various processes by which integrals are obtained 
consist almost entirely in so changing the form of given 
differentials as to make them appear as particular cases of 
the fundamental ones given below. 



Differentials. 



d(v n ) = nv n ~ 1 dv 



dVv = 



2\v 



4r)-A 

\v v z 



d(a v )=Aa v dv 



Integrals. 



(A), .-. 



(B), .-. 



(Bi), .-. 



(E), .". 



V n dv 



V 



n+1 



n + l 



T ,n^ — 1 (a) 



dv 

2Vv 

dv 



=Vv 



v- 



V 



(&) 



(61) 



a v dv = a v /A, 

A = loge a (e) 



d(e v ) = e v dv 



a(log v) = — 
v 



d(sin i>) = cos v dv 



d(cos v) = — sin v dv 



d(tan v) = sec 2 v dv 



d(cot v) = — cosec 2/ y dv 



(F), ••• 



(G), ■•• 



(H), ••• 



(I), 
(J), 



e v dv = e v 



dv , 
— = log v 

v 



cos v dv = sin v 



(/) 
(9) 
(h) 



(K), 



sin v dv = — cos v (i) 
'. sec 2 ?; dv = tan v (j) 

'. cosec 2/ y dv= —cot v (k) 



LOFC 



100 



INFINITESIMAL CALCULUS. 



[Ch. XX. 



Differentials. Integrals. 

d(sec v) = sec v tan v dv (L), /. sec v tan vdv = secv (I) 

d (cosec t v) = — cosec v cot v dv (M), .*. 



d( sin x - ) = 
V a,/ 



dv 



,V 



dUan' 1 - ) = 



V a 2 — v 2 
a d# 



a 2 +v 2 



, / . v \ a dv 

d sec -1 - ) = — 

\ a / Wt) 2 -a 2 
To these may be added: * 
dv 



(N), 
(P), 

(Q), 



cosec v cot i? dv 

= — cosec v (m) 

dv 



Vfi2-,2 



= sin x 



a 



av 1 -v 

-tan -1 — 

a 



a Zj rv^ a 



(n) 



(p) 



dv 
vvv 2 — a 2 a 



:=-sec- 1 -(a) 
a 



1 
I 



Vv 2 ±a' 
dv 



a 2 — v 2 



r=log (v + Vv 2 ±a 2 ), 
1 1 /a + v\ , . 



and 



1 



a — v, 

f a-\-v 



1, 



= 2^ l0g (^j' *l>l a ' 



dv 



1 



log 



v 



vVa 2 ±v 2 a \a+Va 2 ±v 2 
We add the hyperbolic equivalents of (r) ; (s) ; and (t). 

dv . , 1 v 

== = sinh 1 — > 

v 2 + a 2 a 



(r) 



(«) 



(0 



dv 



v 



(r') 



= cosh 1 —i 
Vv 2 -a 2 a i 



* Formulae (r), (s), (0 should be committed to memory with the 
others, as they are of fundamental importance. It will be seen later 
that they may be deduced from the preceding formulae. Compare 

carefully (n) and (r), (p) and (s), (q) and (t). Notice that / 

J v- — a 2 

/dv 
-= s and is therefore known from (s). 
a 2 —v 2 



97.] 



FUNDAMENTAL INTEGRALS. 



101 



and 



f dv 1 , , v , , 

\-z 9 =— tanh 1 —, v\<\a, 

Ja z -v z a a 






1 4. U -1 V Kl 

= — coth l — , v\ > \a, 
a a' ' ■ ' J 



dv 



v V a 2 + v 2 



— sinh x — 
a v 



1 u-i v 

— cosecn x — > 

a a 



v V a 2 — v 2 



— cosh l — = sech 1 — 

a v a a 



(»') 



(O 



CHAPTER XXI. 



FUNDAMENTAL INTEGRALS. II. 



Examples. Formulae (a) to (g). 



• 



1. \ax 3 dx = \ax*, 



2dx 



.r 



I 



(ax 3 + b) dx = \ax* + bx, 

2x~ 2 1 Cdx 1 



= \2x- 3 dx = 



X 2 ' X 2 



X 



2. 
3. 
4. 
5. 
6. 



I 



(#' — a 2 ) 2 dx = (x 4 — 2a 2 x 2 + a 4 ) dx = \x* — %a 2 x 3 + a*x< 



(x 2 -2Y*xdx = i Ux 2 -2)ld(x 2 -2) = \(x 2 -2)§. 

xdx [d(a 2 -x 2 ) ,— 

Va 2 — x 2 J2Va 2 — a: 2 

xdx Cd(ci 2 — Xj 

2 _„2 = ~~i\ „ 2 „2 = ~2 log (a 2 -£ 2 ), by (#). 



r( i-x 2 ) 2 

a; 



c?x 



f 1 



-2x 2 + a; 4 



a; 



dx 



r/i 



X 



2rc-f x 3 )e?.r 



7. \e- 2X dx=-h 



■I 



= log £ — x 2 + ia; 4 . 
<?- 2:r d(-2.r) = -if" 2 *, by (/). 



8. Lre-* 2 d:r= -* 



9. \ax^dx = 2 axh 



e-* 2 d(-x 2 )=-ie-* 2 



-I 



10. 



or - * dx= — 3x _ 3. 



102 



97.] 



FUNDAMENTAL INTEGRALS. 



103 



11. 



13. 



14. 



16. 



18. 



x dx 

a 2 +x 2 



= log vV + x 2 . 12. 



(a + bx) 3 dx = 



(a + bxY 
~46 ' 



V2ax — x 2 (a — x)dx = \{2ax— x 2 )*. 



e ax d x — c ax/ am 
dx 1 



„ j 



do: / 1 

log 



a- x 



a — x, 



s 



(a-xY 3(a-x) r 
x 2 dx _ . / 1 



3 -x 3 



Jlog 



a 3 — a; 3 , 



20 



•I 



dx 



x log X 



= log (logx). 



r 4 — 

22. 3Vxdx = 14. 



'ae dr 



ft 

Cat 
J a 

•i: 

-1} 



17. 



19. 



21. 



23. 



26. 



X ~ 2 7 2 , «X 

— -_e/x = — t=(x + 2). 

xvx vx 



(log x) n 



dx (log x) n+1 
a; n + 1 



a dx = a 2 . 



x"*dx = 2. 24. 



o 

-00 


•1 



X 



dx 



X 



- =1 * 
4 ?' 



6~ a:C Jx = — . 



a 



27. (ax + l)dx = -|a 3 -a-2. 23. e x dx = e-l. 



29. x n dx = 

31 

33. 



w + 1" 

. (2a)* + 1 
(a±x) n ax = — 7- 

J -a n + 1 

00 ^X 1 



30. 



32. 



o 

'2a 
a 



\ // x — adx = %a%. 



" 2 x dx 

4 + x 2 



i log 2. 



34. 



(a + x)" (n-l)(2a)»- 1 ' n> L 
(a + 6x + cx 2 )x dx = T V(6a +'46 + 3c). 



*It is implied in § 92 that a and b are assigned values of x. In 

this example J is to be understood as the limit of / -r when 6 is 

J\ x 
infinite. 



104 



INFINITESIMAL CALCULI'S. 



[Ch. XXI. 



35. 



x dx 1 

o \ o . -> ••> • 

[a*-x 2 y ba- 



37. I" ^^dx = h{log2y. 



39. 
41. 



(a —x) 2 x± dx = yW #-• 



a {a—x^dx 
o \ 2ax— x 2 



= a. 



36. 



' a x dx 



o\ cr+ar 

'00 



= fl(\ 2-1). 



r — 

' Jo V2a-, 



>2rt dx ,— 

40. = 2\^2a. 



42. 



3 3J "Jx = 26/log27. 



CHAPTER XXII. 



FUNDAMENTAL INTEGRALS. III. 



Examples. Formula (h) to (rn). 



•| si 



sin30d0 = * 



sin30d(30) = -J cos 30. 



2. cos 50 cos 30 d0 = ^ 



(cos 80 + cos 20) dd 



1 / sin 80 sin 20 N 



8 



3.* 



d0 



sin cos 
' d0 



5. 



sin 

dd 

cos 



sec 2 ddd 
tan 

d0 



2 sin ^0 cos %0 
' dfo + d) 



d tan 
tan 



log (tan 0). 



sin \d cos \d 



175 = log (tan£0), 



by Ex. 3. 



sin (\n + 0) 



or 



= log tan (in + id), by Ex. 4, 
= log (sec + tan 0).t 



* Integrals 3-11 deserve special attention on account of their fre- 
quent occurrence. 

f This important integral may also be treated as follows: 

f sec- Odd 



i 



sec dd 



J \/tan 2 + 1 



log (sec 0+tan 0), by (r). 



The integrals of Exs. 5, 4, 3 may also be expressed thus (see foot- 
note, p. 36): 



J cos J sin J i 



dO 



X{2d-hn). 



sin z ' v/; J sin cos 6/ 

Numerical valuas of X(6) are given at the end of the book. 



105 



106 



INFINITESIMAL CALCULUS [Ch. XXII. 



6. 



7. 



8. 



9. 



10. 



11. 



tan0d0 = 
cot dd = 
sin 2 ddd = i 

cos 2 0d0=4 
tan 2 0d0 = 



sin Odd 

a~ = — log cos 6 = log sec 0. 

cos ° ° 



cos dd 



= log sin 0. 



sin 
(1 -cos 2d)dd = ±(6-i sin 20). 

(1 + cos 2d)dd = i(d + i sin 20). 

(sec 2 0-l)d0 = tan 0-0. 



sin 2 cos 2 d0 = i sin 2 20 dd = \ 



12. 



13. 



15. 



17. 



18. 



cos d0 
sin 5 



(sin 0)~ 5 d(sin 0) = 



(1 -cos 40)^0 
= i(0-i sin 40). 
1 



4sin 4 0' 



cos(30-l)d0 = J sin (30-1). 14. 



sec 2 40d0 = itan40. 



r sin 3 7/i , 

T7,d0 = itan 4 0. 
u 



COS' 



16. 



cos 2 n0 dd = ^6 + i(sin 2n0)/n. 



d0 



sin + cos 



= iV2l gtan (i7r + i0)=*£V2 J(0-i*). 



Vl + cos dd = 2 V2 sin 40. 



19 



d0 



V2 log tan 1(tt + 0) = V2 A(£0). 



J Vl+cos 
20. J \/l ±sin d0 = 2(sin ^0T cos £0). 



21 



-Ji 



dO 

-f COS 



= tan 40. 



22. 



d0 



l+sin0 



= tan (id -in). 



97.] 



FUNDAMENTAL INTEGRALS. 



107 



23. [sin 50 cos 30 d0 = -J (cos 20 + J cos 80). 

24. J sin 30 sin 20 dO = \ (sin -\ sin 50). 



25. 
26.* 



J 



sin 5 0d0 = (l-cos 2 0) 2 sin 0d0 = -cos + f cos 3 0-i cos 5 0. 



sin 4 dO 



l-cos20\ 2 



d0 



~~ 4 



[1-2 cos 20 + i(l + cos 40)] = f - J sin 20 +& sin 40. 



27. 
28. 
29. 
30. 

32. 
33. 
34. 

36. 

38. 



I 



tan 3 0d0= (sec 2 0-l) tan d0 = i tan 2 + log cos 0. 



sin 3 d0 



(l-cos 2 0)dcos0 



cos 2 
cos 4 0sin 3 0d0 = 
dO 



= sec -f- cos 0. 



cos 2 
cos 4 0(l~cos 2 0) dcos0= -icos 5 0+|cos 7 0. 



sin 2 cos 2 



= tan — cot 0. 31. 



"sin ^0 



sin 



^- d0 = log tan Ktt + 0) 

= K¥). 



fsin 2 d0 

j- = log tan(i?r + iO) - sin = ^(0) -sin 0, 



cos 

d0 
cos 4 



(l+tan 2 0) sec 2 0d0 = tan + Jtan 3 0. 



sin OdO = 2. 



35. cos 0d0 = 0. 



?l* dO 



cos 2 



= 1. 



J 
J 



f*w 



sm 2 0d0 = i7z = 



37. tan 2 0d0 = l-i?r. 



*oob"»*.80. ^™^ = V2-1. 

cos 2 



* Compare the met beds in Exs. 25 and 26 according as the index 
is odd or even. 



108 



INFINITESIMAL CALCULUS. 



[Cii. XXII 



40 



•I 



^ dd 



42. 



In 

u 



tan 



= \ log 2 = '347*. 41. 



fi* 



sec 3 6>tan ddd = 2\. 



qoVO di9 = i(l-log2)=-153. 



j i* 



Jo 



tan 4 0d0 = '119. 
o 



44. 
45. 
46. 



= \ log tan A 7r = " 658 = $ Ki^). 



cos 20 

sec Odd = -521. 



Ml+cos^ /J^ = . 201> 



+ sin 6> 



7T + 2. 



* Use the tables at the end of the book. 



CHAPTER XXIII. 



1. 



dx 



FUNDAMENTAL INTEGRALS. IV. 

Examples. Formula (n) to (/')• 
fir 1 In, 1 , /a + x\ 



a 2 — x 2 ]2aLa + x a — xj ' 2a S 



a — x, 



It also = n - 1 — : \dx = log ( ) . 

\2a[_a + x x-aj 2a to \x-aj 



2. 

3. 
4. 
5. 
6. 

8. 



This is (s). 
dx 



1 



f d(aV3) 1 



V4-3x 2 V3J 



Vl-x 4 ' 

dx 
\ / 2ax — x 2 

dx 



V 2 2 -(xx / S) 2 V3 
d(x 2 ) 



sin -1 ( x 



V3> 



Vl-(o; 2 ) 2 
d(x — a) 



= \ sin _1 (^ 2 )- 



-sin -1 



x — a 



Vx 2 ±2ax 

dx 1 

2ax — x 2 2a 

dx 



Va 2 -(x-a) 2 \ a 

d(x±a) _ , /— — - — , 

= log (x ±a + Vx 2 ±2ax). 

dx 



\ // (x±a) 2 — a'' 
x \ 



log 



X 



\/x 2 — a 2 



\2a — x) ' 
dx 



7. 



x 2 + 2ax 



= ^ log ( 



x 



x 2 + 2a' 



*M- 



rr 



'5. 



X' 



\£ 



1 . la 

~= sin -1 — 

2 a \x 



1 x 

*The integral is also -sec -1 -. These apparently different re- 
sults differ only by a constant (in this case n/2a), and therefore have 
the same differential. 

109 



110 



INFINITESIMAL CALCULUS. [Ch. XXIII. 



In a similar manner deduce (f) from (r) and {V) from (r'). 



9. 



x 2 dx 

l+~xQ = ^ tan~ x . 

x dx 



13 



f x dx . ' (x 2 \ 

f dx 
. — ==: = sec~ 1 e x . 
\Ve 2x ~l 



10. 



12. 



14. 



fx 2 dx 



x 2 dx tl /l+x 3 \ 



do; 



dx 



Vl-e 



= tan _1 6 :r . 



= — sech -1 e*. 



2a: 



15. 



dx 



W 



dx 



_ 1 



~ 2 J(x 2 -x + i)+i 2 
= tan~ 1 (2x-l). 



d(x-*) 



(*-i) 2 + (i) 2 



16. 



17. 



dx 



Vl — x— x 2 

dx 

Vl+x + x 2 



sin - 



= sinh _1 



2x + l \ 

, V5 / 

2x + T 



V, 



dx Vx 2 — a 2 



18. = \/x 2 — a 2 — a sec -1 — . 

J x a 



19. 
20. 
21. 

22. 
23. 
24. 



[Rationalize the numerator.] 

dx \/a 2 — x 2 



x 



dx 



a — x 



= Va 2 — x 2 — a seen -1 — . 

a 



x 



Va 2 — x 2 + a sur -1 — 

a + x a 



dx \x + a x x 

— v = sec -1 — -fcosh" 1 — 

x \ix~a a a 



dx 



x V4:X 2 -9 
dx 



=i sec-^fx), 
1 



\/5x 4 -3x 2 VS 



sec-Mx r- 



x*dx 
V5-4x 3 



d(2xi) 
V5-(2z§) 2 



i- i/ 2 * § \ 



97.] 



FUNDAMENTAL INTEGRALS. 



Ill 



25. 

27. 
28. 
30. 
32. 



x dx 



3 dx f l x dx 

— - = i tan- 1 3 = '624. 26. — = Jtt = -785. 

4 + y.c- J Vl-x 4 



2 + 5 - 2 = T 1 oV / 10tan- 1 (iv / 10) = '318. 

'* x dx , _ B _^ rt f 2 dx 

1 - 4 = ilogl=*128. 29. 

1— x 4 4 & 3 



2 4^1=itan- 1 | = -161. 31. f-^ 

r a; 4 + 4 La; 2 

= i*=-785. 3c 



/•OO 



^ = itan- 1 | = -322. 



! ar\/2a; 2 -l 



rl x* da; 

V8-4x 3 



CHAPTER XXIV. 



INTEGRATION OF RATIONAL FRACTIONS. 

98. An algebraic fraction is rational when it contains no 
surd expressions involving the variable. 

// the fraction is improper, it must first be reduced to a mixed 
quantity. 



Ex. 1. 



x- 



l+x : 



x 2 -l 



l+x 



2* 



' x dx 

7- — 2 = Jx 3 — x + tan-^x. 

1 + x 2 * 



x 5 dx 

l+x 2 



(x 3 -x + J dx = fx 4 - |x 2 + J log(l +x 2 ). 



When the fraction is a proper fraction, it should, in gen- 
eral, be decomposed into partial fractions. See Appendix, 
Note A. 



Examples. 



1. 



x 2 + 3x + l 

x(x-l)(x + 2) 

1 x 2 + 3x + l 

x(x-l)(x + 2) 



± 1_ 



1 1 



1 



2 x 3 x-l 6 x+2 
dx= -ilogx+f log (x-l)-^log (x + 2) 



= ilog 



(*-d { 



! -i 



vV+2x 3 ' 

(l+3x)dx en _1_ ?\,| 

\x~r+x + (l+x) 2 /^ 

= logx-log(l+ a :)- r J-=log (j^) - I |^. 



x +2x 2 +x 3 



112 



98.] 



INTEGRATION OF RATIONAL FRACTIONS. 



113 



L j(*~ 



dx 



4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 
13. 
14., 
15. 



a)(x — b) 
dx 



dx 
a — b 



\x — a x — bl a — b ° \x — bl 



1+2^ 



l+3x + 



2? =1 °s(tt!)- 



CO ^.5 



8x 5 dx 



^ = o; 4 -x 2 + ilog (l+2x 2 ). 
= log (x + 1) 



l+2x 

(2a: + l)dx 



x 



x(x + l)(x + 2) 
dx / x 



(x-f-2) ; 



x(l+x) 2 l0g ' 



dx 



1 



X 4 (l+X 2 ) x 3x 3 

(3x 2 — l)dx 



4-tan _1 x. 



3x4-11 log (x-2)- 2 log (x-1), 



x 2 — 3x + 2 

x 3 + 5x 2 + 8x + 4 0g ( l+x)+ 2+x' 
'(x 3 + l)dx 31 3 . , 

'(1 + x) dx 
x(l+x 2 ) 

2x dx 

1 +x + x 2 + x 3 

2x dx 



tan~ 1 x + log 



x 



Vl+x 2 



= log 



vTT 



X' 



+ tan _1 x. 



1+x 

l+x\* 



(l+x 2 )(3 + x 2 ) log \3 + x 2 ) 

3x + 4 

(x 2 + 3x + 2)~ 



x dx 3x + 4 / x + l \ 

2 + 3x + 2)~ 2 = x 2 -f-3x + 2 + g U + 2/' 



CHAPTER XXV. 



INTEGRATION BY SUBSTITUTION. 



99. To assist in bringing certain differentials under forms 
already considered various substitutions are employed, the 
most important of which will be mentioned in this chapter. 



Ex. 1. 



-5-. Let * = I then dx= - 

ax + bx n z z 



dz dx dz 

-j, and — = 



x 



Substituting, we have 



z n ~ 2 dz log (az 71 - 1 + b) 

+ 6 = -a(n-V) ' by W' 



n-i 



1 / X n ~ 1 \ 1 

lo § h^h^ti-i ) > when z is ^placed by -. 



a(n — 1) & \a + bx n 



x 



2. Making the same substitution and integrating by (a) we 
have 



dx 



x 



3. 

2z dz. 



(a 2 -x 2 )i a\a 2 -x 2 )i' 
dv 



dx 



= ± 



x 



{x 2 ±a 2 )i a 2 (x 2 ±a 2 )V 



Vv 2 ±a' i 



. Let VV ± a 2 = z. Then v 2 ± a 2 = 2 2 and 2v dv = 



dv dz d{y + z) 

z v v-\-z 



'dv 



= log (v + z), 



or 



— =^=r=log (v + Vv 2 ±a 2 ). 
J vr±a 2 

Thus formula (r) is deduced from (g). 



114 



99-103.] INTEGRATION BY SUBSTITUTION. 115 

ioo. Binomial differentials. Any expression of the form 
(ax p + bx q ) r dx, the indices being positive or negative, integers 
or fractions, may be called a binomial differential. For con- 
venience in making the following statements it will be best 
to suppose the binomial differential to be given in the form 

x m (ax n + b) r dx. 

This can be integrated immediately in the following cases: 

(1) When r is a positive integer, expand by the Binomial 
Theorem. 

TYl ~\- 1 

(2) When is a positive integer, let ax n + b = z. 

lb 

Tfb "4" 1 

(3) When Yr is a negative integer, let ax n + b = x n z. 

lb 

io i. When the differential is a function of a + bx let 
a + bx = z n , where n is the L.C.M. of the denominators of 
the indices. 



Ex. 



dx 



(1+X)i+(1+X)a 

dz 



2z dz 

, if l+x = z 2 , 



z 3 + z> 



= 2 



1+2 



= 2 tan- 1 ,? = 2 tan-^Vl+z. 



102. In ■ — . let ax 2 + b = x 2 z 2 . 

(Ax 2 + B)Vax 2 + b 

103. sin m # dd, jra odd and + , let cos 6 = z, 

sin m cos"fl dd, j .-. -sin dd = dz. 

cos m # dd, j m odd and + , let sin 6 = z, 

sin"fl cos m # dd, J .\ cos 6 dd = dz. 

) let tan d = z, 
s'm m ddd, cos m ddd 1 m even and-, .-. cos d=l/(l+z 2 )l, 
sin m dcos n ddd, ?n + n even and-, j sin d = z/(l-{-z 2 )t, 

J dd = dz/(l+z 2 ). 



116 INFINITESIMAL CALCULUS. [Ch. XXV. 

104. More generally, any rational function of sin or cos d 
becomes algebraic and rational when tanJ0 = 2. For 

cos d=(l-z 2 )/(l+z 2 ), sin = 2z/(l+z 2 ), dd = 2dz/(l+z 2 ). 

105. Any rational function of tan 6 becomes algebraic and 
rational when tan Q = z. For dd = dz/{l+z 2 ). 

106. Any rational function of e x becomes algebraic and 
rational when e x = z. For dx = dz/z. 

107. On the other hand, certain algebraic surds are ren- 
dered trigonometric and rational by substitution. For 

if x = a sin d, (a 2 — x 2 )$ = a cos 0; 

if x = a tan 0, (x 2 + a 2 )$ = a sec 6; 

if x = a sec 0, (x 2 —a 2 )? = a tan 6; 

if x = 2a sm 2 0, (2ax—x 2 )% = 2a sin 6 cos d; 

if x = 2a tan 2 # ; (x 2 + 2ax)$ = 2a sec tan 8; 

if x = 2a sec 2 #, {x 2 — 2ax)% = 2a sec tan 0. 

Hyperbolic substitutions may also be employed. For 

if x = a sinh z, (x 2 + a 2 )$ = a cosh z; 

if x = a cosh z, {x 2 —a 2 )% = a sinh z; 

if x = 2a sinh 2 2, (x 2 + 2aa;)* = 2a sinh z cosh 2; 

if x = 2a cosh 2 £, {x 2 — 2ax)$ = 2a sinh z cosh 2. 

108. Since ax 2 + bx + c = — [(2az + 6) 2 + 4ac— 6 2 ], the follow- 
ing general results may be obtained from previous integra- 
tions by the substitution 2ax + b = z. 



(1) 



dx 2 _J 2ax + b 

tan x 



ax 2 + bx + c x/^ac-b 2 \V4ac-b 2 

if \ac—b 2 is + , and 



1 . /2as + 6-V& 2 - -4ac 

log 



Vb 2 -4:ac \2ax + 6 + V 6 2 - 4ac, 



104-109.] INTEGRATION BY SUBSTITUTION, 

if b 2 — 4ac is +. 



117 



(2) 



= =^\og[2ax + b + 2Va(ax 2 + bx + c)l 

J V ax 2 J rbx-\-c V a 



dx 



sin *■ 



2ax— b 



(3) | 



V — ax 2 + bx + c Va \V4ac + ft 2 . 

ris _ 2(2ax + b) 

(ax 2 + bx + c> 3 (4 ac _ 62)Vax 2 + 6a; + c' 



)■ 



(4) 



# da; 



ax 2 + 6x + c 2a 



'(2ax + b)dx—bdx 
ax 2 + bx + c 



(5) 



# dx 



. <»J 



Vax 2 + bx + c 
x dx 



=— log (a£ 2 + 6z + c) — — — x — '- . 

1 / — 7T7 1 — r~ & r ^ 
==— Vaar+os+c— — I- 

c a 2aJ ' 



vax 2 + 6x -f c 



2(te + 2c) 



(az 2 + fcr + c)* (4ac-& 2 )Vaa; 2 + fcr + c' 

dx 



or 



109. If we put x = — in - 

1 . f d£ 

—111 

« J 



£ + & 



(x + ft) Vax 2 + 6x + c 
these integrals will be reduced to § 108 (2). 



Examples. 



1. 



dx 



tan -1 — — 1. Let V2ax — a 2 = z. 
SI a 



_xV2ax~a 2 a 

2. f X -^i5=(io:-l)2v / x + 2tan- 1 v / ^. Let 
J 1+x 

3 f dx 

' }(2+x)\/l+x 



x = z. 



= 2 tair^Vl+z. 



118 



INFINITESIMAL CALCULUS. 



[Ch. XXV. 



4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 
13. 
14. 
15. 

16. 
and 



dx x + a 1 

m (2ax+x 2 )? a 2 \/2ax + x 2 * 



dx 



V(x-a)(x-b) 
dx 



= 2 log (Vx-a + Vx-b). \ 



= 2 sin - 



! \x — a 
\jb~a 



(• Let x — a = z 2 . 



V(x-a)(b-x) 

x\l +x 2 )-2dx = e 1 1 i(7x 2 -2)a+x 2 )l. 

dx V x 2 -1 



xWx 2 -l Sx 3 

dx 1 



(l+2z 2 ). 



(l+x 2 )Vi-x 2 V2 



tan 



— i 



lx 



2x' 



sin 5 # dO=-l cos'O + 1 cos 3 fl - cos 0. 



sin 3 # d# 



cos 4 # 
dd 






cos 3 cos 3 0* 



12. 



d0 



'sin_0 

cos 5 



= -|\/tan 3 0. 



sin 4 cos 2 



f dO 



= -4 cot 3 0-2 cot + tan 0. 



l+sin0 l+tan^fl* 
sin — sin 2 



1 + sin 



d0 



d0 = 20 + cos + 



1+tan W 



a + b cos Va 2 — b 2 
2 



Vb 2 ~ 



tan - 



tanh 



a — b 



\\ja + b 
ib — a 



tan 



w 



b + a 



i°), <*\>\b, 
), a\<\b. 



tan hO 



■ 



17. tan 4 0d0 = itan 3 0-tan + 0. 



109.] INTEGRATION BY SUBSTITUTION. 119 

18. 



dx sec -1 x \/x 2 — l 



19. 



x 3 \/x 2 -l 2 2x 2 

dx 1 , x 

log 



x 



Vao; 2 + 6a; + c \Tc ^ bx + 2c + 2\ / c(ax 2 ~+bx + cj' 



[ dx 1 . , bx — 2c 

20. — ■=— rrsin- 1 - 



a;Vaa; s + 6x — c Vc xV6 2 + 4ac 



CHAPTER XXVI. 



INTEGRATION BY PARTS. 



no. Since v du + u dv = d(uv), .*. v du+\u dv=uv, 

v du, 



u dv = uv— 



or u dv can be integrated provided that v du can be. 
Integration by this formula is called integration by parts. 



Ex. 1. 



2. 



3. 



log x dx= (log x)x —\x e — 

= (log *)*-}<**=* log*-*. 

sin -1 £ dx = (sm~ 1 x)x — x . = x siii- 1 ^ 4- v^l — x 2 . 

J VT^aT 2 



x log x dx 



X 

log x . x dx = (log x)-~- — 



[x 2 ^ dx 
~2 *~& 



x 2 . x 2 

~2 lo g x ~4' 



4. Similarly; 



x n log x dx = , ., log x 



n + l 



(n + l) r 



in. It is often necessary to repeat the integration by 
parts before the complete integral is obtained, 

120 



110-112.1 



INTEGRATION BY PARTS. 



121 



Ex. 

Again, 



x 2 cos x dx = 



x 2 . cos x dx = x 2 sin x — 2 



x sin x dx = 



r 



x sin x dx. 
cos x dx 



Substituting, 



x . sin x dx = — x cos £ + 
= — x cos z-fsin x. 
x 2 cos x dx = x 2 sin # + 2x cos x — 2 sin z. 



ii2. Sometimes the integration reproduces the given ex- 
pression with a new coefficient. 



Ex. 1. Va 2 -x 2 = 



a 2 —x 2 

Va 2 — x 2 



Va 2 —x 2 dx = a 2 



f cfcr 



. va 2 — a; 2 

= a 2 sin -1 

a 



x 2 dx 

. Va 2 — z 2 

a: d( — \^a 2 —x 2 ) 



= a 2 sin -1 — hi^a 2 -^ 2 



a 



va 2 — x 2 c?x. 



Transposing the last term to the left-hand side and dividing 
by 2, we have 



Va 2 —x 2 dx = -^r sin -1 — -frV^-x 2 . 
2 a 2 



a) 



* 



2. Similarly, 



a- 



x 



Vx 2 ±a 2 dx= i-o-log Or + V;r 2 ±a 2 )+-Va; 2 ±a 2 . (2)t 



* (1) and (2) are of frequent occurrence and should be carefully 
noted. They are also easily obtained by the substitutions of § 107. 



t Or, yv 



a 4 



,x . x 



x 2 + a 2 dx=— sinh -1 - — \-—Vx 2J ra 2 , 



i 



// ■ (2 2 XX r 

vr- a 2 dx = — — cosh -1 — ^-Va; 2 - 



a- 



(3) 



122 



INFINITESIMAL CALCULUS. [Ch. XXVL 



3. I see xan-6 dd = tan see 6 - I sec 3 dO t 
whence, since sec*0 = 1 +taa'0 and sec dd = \og (sec 6 +tan 6) 



I 



sec 0tan 2 0d0 = *see 0tan 0-*log (sec + tan0). 



EXAMPLES. 

1 . I x cos j dx = x sin j — cos x. 



2. I tan- 1 



3. 



4. 



o. 



6. 



. . 



~ l x dx = x tan _1 x — los vl + xK 



x tan _1 .r dx = h (1 -\-x 2 ) tan - 1 x— %x. 



x sec -1 x dx = h [x 2 sec _1 x — \ x 2 — 1). 



xe x dx= {x — l)e x . 



z 2 e*dx=(x 2 -2x+2)e* 



e*an .r rf.r = J6*(sin .r — cos x). 



8. | . : an x dx = 2x sin x+ (2— x) 1 cos x. 

9. x sec\r dx = x tan x + log cos J. 

e* sin 2x dx=-£ (sin 2x — 2 cos 2j). 
11. Lr tan\r Jj* = x tan x — log cos x — hx : , 



■ 



112.] INTEGRATION BY PARTS. 123 

f x 3 

12. x 2 (\ogx) 2 dx = ^[(\ogx) 2 -% logrr+f]. 

m 

13. sec 3 dd = i sec tan + \ log (sec + tan 0). 

14. e a * cos wa; dx = -=-; — ,(a cos mx + m sin raz). 



e ax 

a 2 + rri' 



15. 6 a:c sin mx dx = —£— — 2 (a sin mx —m cos mx). 



,2 



16. V^az — x l dx = —^—\ / '2ax — x 2 +—sm- 1 



2 ' 2 V a 

x + a /- a 2 



17. v / 2ax + x 2 rfx = — 7r-V / 2ax + a; 2 — ^-log (x + a + v / 2ax+a: 2 ). 



CHAPTER XXVII. 

SUCCESSIVE REDUCTION. 

113. To integrate sin n # dd, n being a positive integer. 

I sin n # dd = J sin 71-1 ^ sin d dd 

= — sin n_1 (9 cos d + (n— 1) sin" -2 /? cos 2 d dd, 
and writing 1— sin 2 (9 for cos 2 #, 

= -sm n ~ 1 d cos 0+ (n— 1) sin"~ 2 dd— (n- 1) sin"0 dd. 

Transposing the last term and dividing by n, we get 

f . „ ,„ sin n-1 cos d , n—lC . „_ nrx nrt 

sin"0 dd= + sm n " 2 dd. (1) 

J n n J 

Writing n— 2 for n, we have 

f . __ ,_ sin"- 3 0cos0 , n-3f 

sm n-2 dd = - 1 J 

J n—2 n— 2J 

Thus by each integration the index is diminished by 2, 
and hence will in the end depend upon sin0d0= — cos d, 

or d#=0, according as n is odd or even. 



sm n ~ 3 cos d . n— 3f . . n _. 

sm n_4 d0. 



124 



3-115.] SUCCESSIVE REDUCTION. 

By a similar process 

cos n-1 sin . n— 1 



/ 



cos n fl dd = w " • " ""* v + - — ~ cos n - 2 ddd. 
n n . 

cos 2 # + sin 2 # _. fcos 2 0,„. f d# 



f ^ f 
I4 ' Jsm^-J 



sin n 



-d0 = 



Js-m^ + J 



s i n n-2^- 



The first term= Jcos *<*(~ (w _ ^n-ij ) 

__JLL 

*0 n-ljsi 



d# 



-J, 



(w— 1) sin n_ 
d# cos # 



n-2f dd 



ljsi] 



sin n (n— 1) sin" -1 /? n— lj sin n - 2 0' 
r which we may reduce to 

-r-a=logtani0, or j-r-— = — cot (9, 
J sm J sm 2 

cording as n is odd or even. 

dd sin# . n-2f d# 



cos n_2 0' 



Similarly ' jc^ = (n-lTcos-- 1 + ^ : lI; 

ld |c^ = l0gtan(i7r + i ^' |c5^ = tan ^ 



115. [tan*0 d0 = [tan"- 2 (sec 2 6>- l)d0 
= ftan"- 2 d(tan 6) 



tan»" 2 d<9 



_ r tan^fl 
~J n-1 



tan"" 2 d0, 



I 



tan 0d0 = log sec 0, 



<20 = 0. 



125 



(2) 



M'/ 



126 



INFINITESIMAL CALCULUS. [Ch, XXVII. 



f cnt n ~ 1 f) f 

Similarly, cot"0 dd=- _ - cot"" 2 !? dd, 

and [cot 6 dd = log sin 6, [dd = fl. 



n6. 



cos m dsm n ddd = 



/sin n+1 #> 



cos m - 1 0d( — ) 

\ n + 1 I 



cos m-1 # sin n+1 # m— 1 



CO8 m-20 s[ n n+2g fid, 



n + 1 ft + 1. 

and writing sin 714 " 2 /? in the form sin n # (1 — cos 2 #), 
cos m ~ 1 6sm n+1 d m—1 



n + 1 



ft + 1 
m— 1 

'n + 1, 



cos m-2 # sin n # dd 
cos m # sin n # d0. 



Transposing the last term and dividing, 

cos m-1 <9 sin n+1 m— 1 



cos m d sm n 6 d6 = 



m+n 



m+n m 



cos m-20 sm n^ ^ 



In a similar way we might have obtained 

cos m+1 0sin w_1 n— 1 



cos m 0sin n 0d0 = 



m + n m + n_ 

117. The following will present no difficulty*: 



cos m sin n ~ 2 d0. 



cos m 
sin n 

sin n 
cos m 



d0=- 



dd = 



cos m-1 



fti— 1 



(n— 1) sin n_1 ft— 1. 



' cos m ~ 2 
sin"~ 2 ' 



sin 7 *" 1 /? 



ft— 1 



118. 



d0 



(m— 1) cos m *0 7ft— 1 

fcos 2 + sin 2 in 
do 



f sin n_2 
cos m-2 



d0. 



cos m sin n J cos m sin n 

d0 



1 



cos m-2 sin M 



+ 



d0 



cos m sin"~ 2 0" 



11G-119.] 



SUCCESSIVE REDUCTION. 



127 



The first term = 



' 1 d ( -1 \ 

cos m_1 # \(n— I) sin 71-1 !?/ 



1 



(n-- 1) sin 1 

m- 1 f d# 



(n- 1) cos 7 " -1 !? sin 71-1 /? n- 1 J cos m <9 sin 71 " 2 ^ 
Substituting this, we get 



dO 



1 



?n + n-2 



cos^sin 7 ^ (n-l)cos m_1 l9sin 71 - 1 ^ n-1 



dd 



cos m 6s'm n - 2 d' 



By treating the second term in the same way we might 
have obtained 



dd 



1 



ra-hn— 2 



cos m #sin n # (m-l)cos 7n_1 i9sin n - 1 m— 1 



d# 



cos m - 2 #sin n #' 



Hence the integration may be reduced to one of the fol- 
lowing : 



d6, 



dd 



cos 6 sin 0' 



' dd 
sind' 



or 



dd 
cos d 



(Ch. XXII). 



119. The following may be obtained from the preceding re- 
ductions by the substitutions of § 107; they may also be 
obtained directly (cf. § 112). 



1 



x n dx 



x 



n 



x Va 2 —x 2 a 2 (n—l) 



I 



Va 2 



dx 



X' 



n 



Va 2 - 



w 



n 



n-2 



x n 2 dx 



Va 2 - 



x nx/ a 2 - x 2 a 2 (n- l)x n ~ l a 2 (n~ 1)J x n-2^J a 2_ X 2' 



-X' 

dx 



x n dx 



^ _1 V 2ax~ x 2 q(2n— 1) 



V2ax-x 2 n n 

x n dx _x n ~ l Vx 2 + 2ax a(2n— 1) 
V x 2 + 2ax 



n 



n 



x n ~ l dx 
\ / 2ax—x 2 ' 

x n ~ l dx 



I 



n 



n 



(a 2 —x 2 ) 2 dx 



x(a 2 —x 2 ) 2 a 2 n 



n + l 



+ 



n + l 



V x 2 + 2ax 



£-1 



(a 2 — x 2 ) 2 dx. 



128 



INFINITESIMAL CALCULUS. [Ch. XXVII. 



Examples. 

1. Obtain the results of §114 by integrating sec n # dO and 
cosec n # dd. 



2. 



3. 



4. 
5. 



_ x m cos nx mx m ~ l sin nx 
x m sin nx dx= — + — 



n 



n' 

m(m — l) 
n^~ 



x m-2 s [ n nx J x> 



x m cos nx dx 



x m sin nx mx m ~ l cos no: 



n 



n 

n 2 



j 



x m-2 CQS nx ^ 



a: n 6 a2: dx = 
'e ax dx 



x n e ax n 
a a 

e ax 



x n ~ 1 e ax dx. 

[e ax dx 



a 



x> 



(n — l)x n ~ x n — \ 



x n 



-l • 



6. \x m (\ogx) n dx 



£ m+1 (k)g x) n n 



ra + 1 ra + 1 



z m (log£) n-1 dx. 



CHAPTER XXVIII. 



CERTAIN DEFINITE INTEGRALS.* 



120. The first term of § 113 (1) is when # = and also 
when 6=\n) hence 



' 2 • „* j« (n-l)(n-3) . .. 
o n(n— 2)... 



each set of factors being carried to 2 or 1, and a being — 
when n is even, and 1 when n is odd. 



Also, 



cos* 6 dd=- 



s'm n 6 dd. 



121. By examining the results of § 116 it will be found 
that 



f 



mp r,fij* [(m-l)(m-3) ...][(n-l)(n-3) . . .] 
sm m # cos n # dO = - ~ ^7 — — — t^t 1 ■ «, 

(m + n)(?n + ?i— 2) ... 



7T 



each set of factors being carried to 2 or 1, a being — when 

Zt 

m and n are both even, and 1 in all other cases. 
This reduces to 



if n=l, and to — if m==l. 



ra+1 n+1 

122. Many integrals may be reduced to the foregoing. 



Ex. 1. 



a x n dx 



o^a 2 — x 2 



= a n 



sin*fl dd. Let x = asind (§ 107). 



* For a collection of indefinite and definite integrals see Peirce's 
Short Table of Integrals (Ginn & Co.). 

129 



130 



INFINITESIMAL CALCULUS. [Ch. XXVIII 



Jo 



x 2 ) dx = a n+l 



cos n +^ dd. 



o 



3. ) x m (a n --x 2 )~dx = a m + n + i \ sm m d cos n+1 d dd. 
Jo Jo 

■K 

' 2a x n dx 



■ 

Jo 



V2 



= 2(2a> 



ax — X' 



sin 2n dd. 



o 



5. 



6. 



2a 



x rn (2ax-x 2 )-dx = 2(2a) m + n + 1 \ sm 2m + n+l d cos^ddO. 



o 

" x dx 



A si 
J o 







(a 2 -^x 2 ) 



n a n-i 







cos»~ 2 # dd. 



a -z 

7. x m {a-x) n dx = 2a rn + n + l \ si 
Jo Jo 



sin 2 ™* 1 /? cos 2n + l 0J0. 



123. From § 93 it is plain that 

'b 



. 



}(x) dx=— f(x) dx; 
b 



that is, interchanging the limits merely changes the sign 
of the definite integral. 

124. It is possible in certain cases to arrive at the value 
of a definite integral when the indefinite integral is unknown. 
The following important integral is an illustration. 



roc 



To prove 



.-r2 



dx 



\ K 



2 ' 

From § 120 we have 



7T 

f T ■ 

sin" 

Jo 



Odd . 



i 



- 



sm» +1 ddd = 
tt + 1 2 



(1) 



123, 124.] CERTAIN DEFINITE INTEGRALS. 



131 



Let s'm n d = e~ x2 , or # = sin -1 



(,-). 



/. dd= 



2X n | 

n Ix dx 



SJl-e n 



n ^_- 
\Je » -1 



which, by the Exponential Theorem. 



Substituting in (1) we have 



f° 2 gj* rf.r |° 12 e 'dx 



-12 



or 



i 



n /n + h 



*„-f 2 



e J ~ dx 



X 



-*=(— ) 

\ n / 
IT" 



rfx _ n - 
~n+14" 



Now let r»=oo : then D = 1. and — — [ = H — ) also- 1, 

n \ - n/ 



- q: 



e -*"" rfx 



pot 



4 ' 



Examples. 



e -**dx= 



Jo 



\ - 



1. *wa*0d0=2\ 2 wi*0d0 3 n>0. 

Jo Jo 

2. cos n dd = 0, n odd, and =2 2 cos n d dO, n even 
Jo Jo 

x sin x cos rix dx = ( -l) w+1 : _, ; n an integer not 1, 



132 



INFINITESIMAL CALCULUS. [Ch. XXVIII. 



and 



x 



= — 7 if n = l 
4 



f * • i / i \ nn 

4. x cos a; sin nx ax = ( — l) n 2 _i > n an integer. not 1, 



and 



7T 



— -T- if n=l. 
4 



5. The Gamma Function. The integral 



'00 



x n - 1 e~ x dx (n positive 



is called the gamma function and is represented by T(n). Inte- 
grate by parts * and show that 

r(n)=-r(ri + l), or r(n + l) = nr(n). 
n 

6 Show that T(l) = 1, and deduce T(2) = 1 T(3) = 1.2, T(4) = 
1.2.3,... r(n) = (n — 1)! if n is an integer. 

7. Show that r(^) = v / 7r. Let x = z 2 in the integral. Deduce 
/*( n + £) = l .3.5... (2n-l)V^/2 n , n an integer. 

r(n) 



8. 



/•CO 



00 



x n-i e -arc ^ = — ^-. | n and a> 



9. x 2n - l er x2 dx = %r{n), n> 0. Let x 2 = £ 
Jo 



cfc = r(n), n> 0. Let # = e~ z . 



io -£( io ^) n_1 

11. I a*- 1 (log -J dz =— ,mandn>0, 



* £ T x n e- x = for all values of n. 



»X = 00' 



For, x n €r x = 



1 \n 



n 



1 - 



iX 



x n 2n 2 



-0 if n>0. 



The conclusion is obvious directly if n<0. 



124.] CERTAIN DEFINITE INTEGRALS. 133 

Many other integrals may be expressed by means of gamma 
functions. The following, known as the Beta Function, is an 
important illustration (see Williamson's Integral Calculus, § 121). 



I 



r (tyi) r (ti) 

x m-i(i _ x )n -i^ = v . V > m and n> 0. 

q 1 \JfYl ~T"Tl) 



Assuming this result deduce : 

ca r(m) r(n) 

12. x m - 1 (a-z) n - i dx=a m + n - 1 r \ , / . 
J I (m + n) 

r«> a 7 *- 1 ^ r(m) r(n) 

J (l+z)™+"~ r( m + n) * 

/- W*) 

„, f 1 d£ Vtt W 

14. 

Jo 



r{m) r{n) 1 

Let l+x=— • 
z 



Vl-x n n r 

' 2 



7T 



Let sin 2 0=z. 



/m + n + 2\ 
\ ~2~ I 



15. f sm m cos n ddd = - , wandn> -1. 

2 „ /m + n -f 2\ 
J o 



i. J 2 sin"0 cW = I 2 cos«<? <«= _ — 

2 rg + i) 



^2 

Values of r(n) for values of n between 1 and 2 are given at 
the end of this book. Other values are unnecessary on account 
of the relation r(n + l)=n r(n). 



CHAPTER XXIX. 



AREAS AND LENGTHS OF PLANE CURVES. 
SURFACES AND VOLUMES OF SOLIDS OF REVOLUTION. 

125. Let P be a point on a continuous curve CD whose 
equation is given. Let OA = a. OB = b. OM = x, MP = y, 

MX=dx, RQ = Jy. RT = dy.. 
We have seen . §§ 94. 95) that 

(1) The area ABDC = I ydx. 




* 



(2) The volume formed by the 
revolution of this area about 



OX=7Z 



M N 



5 X 



\r dx. 



Fig 



Let the length of the curve 
measured from some point up 
to~P be s; then PQ = Js, PT = ds. Also (§§ 92. 93) £IJs = 
£2ds. Hence 



rx=b 



The length CD= I ds. where ds = \ dx 2 -dy 2 . 

J x = a 

Js and ds being equiva'ent infinitesimal we assume that 
they form equivalent areas in revolving about the .r-axis. 

* lhe area = sin w / ydx if the angle between the axes is oj. 

J a 

134 



125.] 



AREAS, LENGTHS, SURFACES, VOLUMES. 



135 



But ds describes the lateral surface of the frustum of a 
cone whose area 

= 2-0/ + \dy)ds = 2-y ds + 1, 

where / is a higher infinitesimal. Hence 

(4) The area of the surface formed when CD revolves 



about OX is 2- 



rx = b 



y ds. 

x = a p 

Since higher infinitesimals 
are to be omitted in finding 
the element of an integral we h 
may use dy for Ay and accord- G 
ingly regard dx and dy as 

simultaneous infinitesimal in- 

crements of x and y in the 
same figure. Thus (Fig. 67) 

roF 




A M N 

Fig. 67. 



(5) The area ECDF= xdy, and the volume formed by 



OE 



rOF 
revolving this area about OF = -| x 2 dy. 



If CABD revolves about OY, the volume formed by MQ = 
y dx . 2zx + /. Hence 

(6) The volume formed by revolving CABD about 

•6 

OY = 2- xy dx. 



(7) Similarly the volume formed by revolving CABD 
about BD 

= y dx .2r:(b—x) = 2z\ (b—x)ydx. 
Other relations may be written down in a similar way. 



136 



INFINITESIMAL CALCULUS. 



[Ch. XXIX. 



Ex. 1. To find the length of the semi-cubical parabola ay 2 = x 3 
(Fig. 68) from the origin to the point (a, a). 

3 
From the equation we have dy= — -=x%dx % 

2v^ 




' rr-<> — r, ^a + 9x 
ds = Vdx 2 + dy 2 = — : — ^r— dx. 



OD = 



i 



2Va 

2\Ta L 27Va -J 

13^13-8 



27 



a. 



2. The area OBD = %a\ 

3. The volume of OBD about OX = \na\ 






i < 






" OY=±xa 3 . 
11 BD = &m\ 



4. Find the surface of revolution of the cubical parabola a 2 y = x* 
about OX, x varying from to a. Arts. ~;(10\/l0-l)a 2 . 

126. It is sometimes desirable to express both x and y in 
terms of a third variable. 

Ex. The equation x$+y$ = a$ (Fig. 18) is satisfied if we put 
x = a sin 3 0, y = a eos 3 #. 

Then dx = 3a sin 2 cos 6 dd, / . ydx = 3a 2 cos 4 sin 2 dd f 

Jo Jo 

which, § 121, 



Sa< 



6.4.22" 32 



na 2 . 



,\ the whole area bounded by the curve = |7ra 2 . 
For the length, ds = VoV + dy 2 = 3a sin 6 cos 6 dd, 

f ** . 
.'. whole length = 12a sin 6 cos d# = 6a. 



Similarly it may be shown that the volume of the solid made 
by revolving the whole area about one of the axes = rVk^a 3 , and 
that the surface of this solid = J t 2 -7:a 2 . 



126-129.] AREAS, LENGTHS, SURFACES, VOLUMES. 137 



127. It will often be necessary to determine the limits 
of the integration from the equation 
of the curve. Thus in finding the 
whole area enclosed by the curve 

a 2 y 2 =x 2 (a 2 — x 2 ), 

it will be seen that the curve cuts 
the x-axis at (±a, 0) and that the 
general shape is that of Fig. 69. 
Hence the complete area 




Fig. 69. 



ra 

= 4 \ y dx = 



#a 2 . 



The volume of the solid of revolution about the a>axis 
= tVt& 3 > and about the 2/-axis = |7r 2 a 3 . 

128. When y is negative the sign of ydx is — , and accord- 
ingly an area lying below the axis of x will be affected by 

the same sign. Hence in calculating 
an area, care must be taken that y 
does not change sign between the 
given limits. Thus in the curve 

y = x(x-l)(x-2), Fig. 70, 

y is + from # = to x=l, — from 
x=l to x = 2; it will be found that 




n 



y dx 



= J, I ydx=-l, 



y dx = Q, 



And generally the sum-limit given by a definite integral 

f(x)dx is that of the algebraical sum of the elements, 
1 

which will be equal to that of the arithmetical sum only 
when f(x) is of the same sign for all values of x between a 
and 6. 

129. If y is infinite in a given interval of x, the area will 
have a limit (and will therefore remain finite) if the indefi- 



138 



INFINITESIMAL CALCULUS. 



[Ch. XXIX. 



nite integral ydx remains finite for the interval of x in 

question. For example, if 2/ 3 (x-l) 2 =l, Fig. 71, j/=oo 
when x = 1 , and 




Fig. 72. 



o x 

Fig. 73. 



Thus if the area is imagined as described by an ordinate 
which starts from the ?/-axis and moves towards the asymp- 
tote x=l, the area = 3, and this is what is meant by the 
area between the curve, the axes, and the asymptote. Simi- 
larly if the ordinate starts from the line x = 2 and moves 
towards the asymptote x=l the area =3, and the sum 

'2 



of the area-limits = 6 = 



y dx as if y were continuous for 



o 



the interval [0, 2] of x. 

Similarly if y*(x- 1) = 1, Fig. 72, 



y dx 



L J o Ji Jo 



the algebraical sum of the area-limits. 

But if y(x— 1) 2 = 1, Fig. 73, the indefinite integral 



y dx= 



1-x 



= oo when x£l, and the area=oo. In this case 



130] AREAS, LENGTHS, SURFACES, VOLUMES. 139 

2 

ydx= — 2, which represents no part of the area for the 
o 

interval [0, 2] of x. On the other hand, the area for 
[2, oo]=l. 

130. As x increases the volume-element ny 2 dx changes 
sign only with y 2 , i.e., when y becomes imaginary. Thus in 
Fig. 70, 

2 16tT f 1 

ny 2 dx = — ^ = 2 ny 2 dx. 
1U5 Jo 



}; 



Examples. 

1. The circle x 2 + y 2 = a 2 . Show that 

(1) Area = 7ra 2 . 

(2) Length = 2na. 

(3) Volume of sphere = f na 3 . 

(4) Surface of sphere = 4;ra 2 . 

2. The witch y 2 {a — x) = a 2 x, Fig. 54. 
Let x = a sin 2 #, then y =a tan 6. 

(1) Area between curve and asymptote = -ma 2 . 

(2) Volume of this about asymptote = ^n 2 a 3 , 

(3) Volume of same area about OF = |^ 2 a 3 . 

3. The cissoid y 2 (a — x)=x 3 , Fig. 41. 
Let x = a sin 2 #, then y = a sin 2 # tan 6. 

(1) Area between the curve and asymptote = \na 2 . 

(2) Volume of this about asymptote = \n 2 a 3 . 

(3) Volume of same area about OY =\iz 2 a 3 . 

4. Find the area bounded by the rectangular hyperbola xy=l, 
and the lines y = } x = l, x = n. Ans. log n. 

5. The curve y 2 (a 2 — x 2 ) = a 4 , or x = asm 0, y = a sec 6. 

(1) Area between curve, t/-axis, and asymptote x = a is xa 2 . 

(2) Volume of this about 2/-axis = 4^a 3 . 

(3) Volume of same area about asymptote = 2na z {n— 2). 

6. The curve y = e~ x . 

(1) Area from x = to x = oo is 1. 

(2) Volume of this about x-axis = ^7r. 

(3) Convex surface of this solid = tt[V2+ log (1 + V2)]. 



140 INFINITESIMAL CALCULUS. [Ch. XXIX. 

7. The curve x 2 y 2 + a 2 y 2 = a 2 x 2 . 

The area between the curve and each asymptote = 2a 2 . 

8. Find the area between the following curves and the a>axis: 

(1) {y-x) 2 = x\ Fig. 30. Ans. A. 

(2) {y-x 2 ) 2 = x\Yig. 31. A. 

(3) a 2 y = x(x 2 -a 2 ) } Fig. 17. \a 2 . 

(4) y(l+x 2 ) = l. 

(5) y = x(l-x 2 ). i. 

(6) ?/ = a; 2 (z-l). 



i_ 

12* 



4 
TITS"- 



9. Find the area of a loop of the curves: 

(1) y 2 = x*{2x + \), Fig. 34. Ans. 

(2) y 2 = x 2 (2x + l), Fig. 32. A. 

(3) ay 2 =(x — a){x — 2a) 2 . j\a 2 . 

10. The parabola (-) + (-y =1. See Ex. 11, p. 91. 

(1) Area between curve and axes = ^a&. 

(2) Volume of this about OX=j^ab 2 . 

11. The cycloid x = a(6— sin 6) } y = a(l — cos 6), Fig. 19. For a 
single arch: 

(1) Area = 37ra 2 . 

(2) Length = Sa. 

(3) Volume about base = 5^ 2 a 3 , 

(4) Surface of this solid = -\^-7ia 2 . 

(5) Volume of the area 37ra 2 about tangent at vertex = 77r 2 a s . 

(6) Show that in Fig. 20, s 2 = 8ax (s = OP, x = OM). 

12. The curve # = a(l-cos 0), y = ad; Fig. 20. 

(1) Area = 27ra 2 . 

(2) Volume of this about OX = n(n 2 -4:)a\ 

(3) Volume about OY==5n 2 aK 

x 2 y 2 

13. The ellipse — + — = 1, or x = a sin 6. y = b cos 6. Show that 

^ a 2 b 2 

(1) Area = 7ra?>. 

(2) Volume of prolate spheroid * = %izab 2 . 

* The solid formed by the revolution of an ellipse about its major 
axis. 



130.] 



AREAS, LENGTHS, SURFACES, VOLUMES. 



141 



27tab 
(3) Surface of prolate spheroid = 2nb 2 + sin -1 e. 



(4) Volume of oblate spheroid * = §7ra 2 6. 

(5) Surface of oblate spheroid = 2na 2 + — log (- J . 



Note. — The eccentricity e = Va 2 — b 2 /a. 

x 2 y 2 
14. The hyperbola — 2 — r- = 1, or x = a sec 6, y = b tan 6. 

Show that the area bounded by the curve, the z-axis, and 
the ordinate at the point (x 1} y x ) is 



i^i-ia&log (~ 1+ ^7 > 



and hence that the second term in this result is the area of the 
hyperbolic sector OAP, where is the centre, A the vertex, and 
P the point on the curve. 

15. The parabola y l = lax 1 Fig. 74. 

If OA=x lf AB = y ly show that 

(1) Area 0AB = %x 1 y 1 . 

(2) Length OB = ^ Via 2 + y 2 

4a 




+ a log Vi+^+Jh*. 
2a 

(3) Volume of OAB about OX = \ny l 2 x l . 

(4) Surface of this solid = ^ [(4a 2 + y l 2 )* -8a 3 ] 

3a 

= — [normal 3 — subnormal 3 ]. 

(5) Volume of OAB about A5=A?rx 1 2 i/ 1 . 

(6) Volume of OBC about OY =\izx 2 y x . 

(7) Volume of OBC about BC = iny l 2 x l . 



* The solid formed by the revolution of an ellipse about its minor 

axis. 



CHAPTER XXX. 



SIMPSON'S RULE. VOLUMES FROM PARALLEL SECTIONS. 
THE PRISMOIDAL FORMULA. LENGTH OF A CURVE 
IN SPACE. 

131. Simpson's rule. An area (Fig. 75) is bounded by 
a line which is taken as the z-axis, a curve, and two ordi- 




nates of length yi, y 3 , at a distance h apart, and y 2 is the 
ordinate midway between them. 
The area 

A = ^h(y 1 +4y 2 + ys), (1) 

# 

provided that the equation of the curve is of the form 

y = a+bx + cx 2 +dx 3 , (2) 

where a, b, c, and d are constants. (1) is the statement 
of Simpson's Rule. 

For convenience take the origin at ; the foot of the middle 
ordinate. Then the area 



A 



y dx= (a + bx + ex 2 + dx s )dx 

— \h J — i;h 



= ah+ xV c ^ 3 = $h (6a + \ch 2 ) . 
V\, V2> 2/3 are the values of y in (2) when x=~ \ln, 0, \ln\ 
•'• 2/i+2/3 = 2a + icA 2 , and y 2 =a. Hence (1). 



142 



131-133.] 



SIMPSON'S RULE. 



143 



The origin may be any point in OX, for the equation 
would remain of the form (1) if the origin were transferred 
to 0. 

132. When the equation is not of the form (2), or is 
altogether unknown, the area may be divided into four, 
six, or an even number n of parts by equidistant ordinates, 
and (1) applied to each part; the result will be a more 
or less close approximation to the correct area. This de- 
pends upon the fact that y can, in general, be expressed 
as a series of powers of x, and that higher powers than the 
third may, for purposes of approximation, be neglected if x 
is small. Formula (1) now becomes 

h 
5- [2/1+4(2/2+2/4 + . • .)+2(2/3+2/5 + . • • ) + 2/n+lL 

Oft , 

h being the whole base, n the number of parts, y 1 and y n+1 
the extreme ordinates, y2, 2/4? • • • the even-numbered ordi- 
nates, y s , 2/5? • • • the remaining ordinates. 

Ex. If in Fig. 75 the base h were divided into three equal parts, 
show that the area 

= ih(y 1 +3y 2 + 3ys + y 4 ) *, 

where y x and y 4 are the extreme ordinates, y 2 and y 3 the inter- 
mediate ones. 




Fig. 76. 



133. Volumes from parallel sections. Let a solid be cut 
by parallel planes at perpendicular distances a, x, x-\-dx, b 



* Another of Simpson's Formulae, 



144 



INFINITESIMAL CALCULUS. 



[Ch XXX. 



from a fixed point, and let A be the area of the section at 
distance x. Then if A can be expressed as a function of x, 
the volume of the solid between the extreme planes is 



J a 



A dx. 



For the volume of the slice of thickness dx is {A+i)dx, 
where i is infinitesimal, .'. the element of the integral is 
Adx. 

2 2 2 

Ex. 1. To find the volume of the ellipsoid — ,+^+-^ = 1. The 

a 2 b 2 c 2 



equation may be written 




V 



+ 



= 1, 



which, x being regarded as con- 
stant, is the equation of a 
section at a distance x from 
the origin. The area of any 
ellipse y 2 /a 2 + z 2 /fi 2 = l is 7m/?. 
Hence the area of the section 
of which DEF is a quadrant is 



Fig. 77. 



1-* 



H'o^) 



nbc 

volume is 
^9) dx = ^7zabc. 



x 2 \ 
a 2 )' 



the whole 



2. Find the volume of the elliptic paraboloid y 2 /b 2 -\-z 2 /c 2 = 2x 
from x = to x = a. Ans. na 2 bc. 

3. Find the volume enclosed by the plane x = h and the surface 
(1) y 2 /x 2 + z 2 /a 2 = l, (2) xy 2 + az 2 = ax 2 . 

Ans. (1) \Ttah 2 , (2) %7za*hl. 

4. Find the volume of the tetrahedron formed by the cooidi- 



X V z 
nate planes and the plane " +7- H — = 1. 
r a c 



Ans. \abc. 



133.] 



VOLUMES FROM PARALLEL SECTIONS. 



145 



5. Two cylinders of altitude h have one extremity, viz., a circle 
of radius a, in common; the opposite extremities touch each 
other. To find the common volume. 

A section of the common volume parallel to the plane CDEF 
(which contains the centres of the circles) and at a distance OA =x 
from that plane is a triangle GBH similar to EQF. The area of 
EQF is ah. 



GBH AH 2 



a 2 — x 2 



ah OF' 



a' 



GBH=-(a 2 -x 2 ). 
a 



a 



a 



(a 2 ~x 2 ) dx = ^-a 2 h. 





Fig. 78. 

6. A square moves with the middle points of its sides on the 
circumferences of two equal circles at right angles to each other. 
To show that the volume and surface of the groin thus formed 
are each 4A times those of the inscribed sphere. 

Let BCDE (Fig. 79) be one position of the square, OA = x y AP = y. 
The volume and surface elements of the sphere are ny 2 dx, 2nyds; 
those of the given solid are (2y) 2 dx, 4o(2y)ds; hence the proposi- 
tion. The volume and surface are therefore ^fa 3 , 16a 2 . 

The solid is evidently the common part of two equal right 
circular cylinders whose axes intersect at right angles. 

7. A right circular cylinder is sharpened to an edge coinciding 
with a diameter, the equal plane faces forming a wedge. Find 
the volume cut off. 



146 



INFINITESIMAL CALCULUS. 



[Ch. XXX. 



Let a length h be cut from opposite sides of the cylinder of 
radius a. Sections may be made by planes parallel to the axis 
and the diameter, or parallel to the axis and perpendicular to the 
diameter. Ans. ±a 2 h. 

Show that for any diameter of a right elliptic cylinder the 
result is 4 abh. 

8. A parallelogram moves with its angular points on two ellipses 
which have a common axis. The semi-axes are a, b, c, and the 
angle between the curves is co Show that the volume is \abc sin co. 

9. Show that the volume of any cone or pyramid = J base X 
height, assuming that the area of a section parallel to the base 
varies as the square of its distance from the vertex. 

10. A straight line is parallel to a plane which contains a closed 
curve. Another straight line moves so as to intersect the curve 
and the fixed straight line and remain perpendicular to the 
latter. Show that the volume of the right conoid thus formed = 
% base X height. 





Fig. 80. 



Fig. 81. 



11. Form of an inverted column of uniform strength. Let A be 
area of a horizontal section at a distance x above the base, which 
is also assumed to be horizontal and of area a. The prescribed 
condition is that A varies as the volume V below A ; hence dA 
varies as dV. 

.*.dA=kAdx, or dA/A=hdx. 
Integrating, logA=kx + c. But A= a when x = 0, .'. loga = c. 



.'. log (A/a) = kx, or A 



= ae kx 



12c Such a column is to be cast in the form of a solid of revo 



134.] THE PRISMOIDAL FORMULA. 147 

lution, R and r being the radii of the extremities, and h the height. 

nh(R 2 — r 2 ) 
How much metal is required? Arts. Vol. = ^- 



M") 



134. The prismoidal formula. The extremities of a 
solid are parallel planes of area A 1, As, at a distance* h apart, 
and A 2 is the area of a parallel section midway between 
them. The volume 

v = ih(Ai+±A2+A s ), (1) 

provided that the area A of any section parallel to the ex- 
tremities can be expressed in the form 

a + bx+cx 2 +dx s , (2) 

where x is the distance of the section from a fixed point. 



(1) is the Prismoidal Formula. Since the volume 



Ji. ClXj 



the proof is the same as for Simpson's Rule. The Pris- 
moidal Formula will give exact values of the volume of 
many of the common solids, such as cones, pyramids, prisms, 
spheres, ellipsoids, paraboloids, etc. It will apply to Exs. 1-9 
of § 133. (In Ex. 7 it will apply to the second ' mentioned 
sections, but not to the first.) 

Ex. 1. The area of a section of a sphere at a distance x from the 
centre is n(a 2 — x 2 ), which is of the form (2), hence the Prismoidal 
Formula will apply. For the whole volume h = 2a, A 1 =A3 = 0, 
A 2 = 7ia 2 . ,\ V = %na§. 

2. Find the volume of the greatest solid that can be cut from 
a sphere of radius a, the parallel sections to be regular polygons 
of n sides. Arts. %na 3 sin 27r/n. 

The volume of a sphere may be deduced. For 

o o . « / a sin27r/n . „ _ 
$na 3 sin 2^/n=|^a 3 — - — - — = |^a 3 when n = oo . 

2n/n 



148 



INFINITESIMAL CALCULUS. 



[Ch. XXX. 



135. Length of a curve in space. 

Ex. 1. To find the length of the curve of intersection of az = x 2 
and 3a 2 y = 2x* from the origin to the point (x lf y lt z x ). 

(2x\ 2 ' 
1+— J dx\ 



s = 



'*i / 2x 2 

\ a 



2 x x z 
lx = x l +^—£ = x 1 + y 1 . 



2. Find the length of the helix x = a sin nz, y = a cos n&, from 
the origin to the point (x ly y u zj. Ans. z 1 ^l-\-n 2 a 2 . 



CHAPTER XXXI. 



POLAR COORDINATES. 



136. Let be the pole or polar origin, OA the polar axis 
or initial line, (#, r), (d + Jd, r+Jr) the coordinates of P 
and Q, ([> the angle which the tangent at P makes with the 
radius vector OP. Take PR perpendicular to OQ. Then <J> 
is the limit of OQP as Q approaches coincidence with P, 
and teaiOQP=PR/RQ. 





Fig. 82. 



But PR = rsinje = rJd + I lf (§16), and 

RQ = r + Jr— r cos J0 = Jr+r(l — cos Jd) = Jr + I 2 . 

{•*• •! 1 • f W/l/ . Li/I 

Similarly sinf = r^ ; cos = — . 



ds 



149 



150 INFINITESIMAL CALCULUS. [Ch. XXXL 

Squaring and adding, 1 = (r 2 dd 2 +dr 2 )/ds 2 7 
or ds 2 = r 2 dd 2 + dr 2 . 

137. Through the origin (Fig. 83) let a line be drawn 
perpendicular to the radius vector OP, meeting the tangent 
in T and the normal in N. Then TP is called the polar tan- 
gent, NP the polar normal, TO the polar subtangent, and 
ON the polar subnormal. The lengths of these lines in terms 
of r and S can be written down at once; e.g., 

TO = r tan <l> = r 2 dd /dr. 

The tangent at any point P is easily drawn by calculating 

OT and then joining T to P. 

138. Some of these quantities are more conveniently ex- 
pressed in terms of d and the reciprocal of r. Calling this u, 
we have u=l/r, du= — dr/r 2 ; hence the polar subtangent 

TO=~d6/du. 



The polar coordinates of T are [hn + d, - 



dd\ 
dul ' 

Let OG, the perpendicular on the tangent, =p. 

Then v OTP is a right-angled triangle and OG the per- 
pendicular from the right angle to the hypotenuse, we have 

1 _1_ J_ J_ 2 AM 2 ■ m 

OG 2 OP 2 ~^OT 2 ' or p2 ^ + W ' W 

139. The polar equations of some of the commoner curves 
are as follows: 

(1) r cos 6 = a, a straight line. 

(2) r = a cos #, a circle of diameter a (origin a point on 
the circumference, initial line a diameter). 

(3) r 2 cos 26 = a 2 , a rectangular hyperbola, Fig. 98 (origin 
the centre, initial line the transverse axis). 

(4) r 2 =*a 2 cos 20, a lemniscate, Fig. 27 (origin the centre, 
initial line the axis). 



137-139.] 



POLAR COORDINATES. 



151 



(5) H cos \d= a$, or r(l+cos 6) = 2a, a parabola (origin 
the focus, initial line the axis). 

(6) r*=a* cos £0, or r=^a(l+cos d), a cardioid, Fig. 89. 

(7) r(l+e cos 6) = m, an ellipse, hyperbola, or parabola 
according as the eccentricity e<, =, or >1 (pole the focus, 
initial line the axis, m half the latus rectum). 

(8) r = n(l+e cos 0), a limagon, Figs. 88, 89, 90, according 
as e<, =, or >1. 




Fig. 84. 



(9) r = ad, a spiral of Archimedes, Fig. 84. 
(In Figs. 84, 85, 6 varies from a little less than —2k to 
a little more than 2tt). 




Fig. 85. 
(10) rd = a, a reciprocal or hyperbolic spiral, Fig. 85. 




Fig. 86. 



(11) r 2 d = a 2 , a lituus, Fig. 86 {6 is necessarily +, and 
varies in the figure from to a little more than 2tt, r is ± 
for a given value of 6). 



152 



INFINITESIMAL CALCULUS. 



[Ch. XXXI 



(12) 



r = a d , a logarithmic or equiangular spiral, Fig. 87 




r = a when 
when 6 



is 



Fig. 87. 



(r=l when # = 0, 
6 = 1 radian, r< 1 
negative). 

Since 6 may be supposed to 
increase or decrease without 
bound, each spiral ^consists of 
an infinite number of whorls or 
spires. 

140. Equations (1) to (6) are all included under the form 
r m cos m6 = a m ; in (1), (3), and (5) m has the values 1, 2, ^, 
respectively; in (2), (4), (6), it has the values — 1, —2 
— J. In all cases a is the intercept on the initial line. The 
equation r m sin md = a m represents the same series of curves, 
the initial line having been turned backward through the 
angle n/{2m). Similarly (9), (10), (11) are particular cases 
of the equation r m = a m 6 n . 

141. The radius vector of the limagon, equation (8), is 
proportional to the reciprocal of the radius vector of a conic 
section, equation 7; hence the limagon is called the inverse 
of a conic section with regard to a focus. Since r = en cos + n, 
the radius vector is equal to that of a circle of diameter en 
plus a constant line n, and hence the curve is easily con- 
structed. (The construction or auxiliary circles are shown 
in the figures.) 






Fig. 88. 



Fig. 89. 



Fig. £0. 



When 6 = 1 the curve becomes a cardioid (eqn. 6), which is 
therefore the inverse of a parabola. When e = 2 the curve is 



140-142.] POLAR COORDINATES. 153 

called a trisectrix, the loop then passing through the centre 
of the circle. 

Examples. 

71% 

1. If r m = a m 6n i show that tan <p = -d. 

n 

(Differentiate logarithmically.) 

2. If r m cos ra# = a m , or r m = a m cos md, show that tan </> = cot m#, 
i.e., that the angle between the radius vector and normal = md, 
and hence that GO A (Fig. 83) = (m-l)0. 

3. In the logarithmic spiral r = a° show that $> is constant and 
= cot -1 (loge a). 

In Fig. 87. a = 1*318 cm.; show that ^ = 74° 33'. 

4. To find the polar subtangent of a conic. 

From the equation 1 +e cos = m/r = mu we have — e sin 6 dd 
= mdu, and the polar subtangent = — dd/du = m/(e sin 6). 

5. In any conic prove that 

1 J2/1 l-e 2 \ 

p 2 m\r 2m / ' 

6. In the curve r m cos md = a m prove that pr m - 1 = a m . 

7. Changing the sign of m, show that pa m = r m+1 in the curve 
r m = a m cosmd. 

8. Show that the polar subnormal of any curve = dr/dd. 
In what curve is the polar subnormal constant? 

9. In what curve is the polar subtangent constant? 

10. Show that the polar normal = ds/dQ. 

Asymptotes. 

142. The position of any line is known when its direc- 
tion and one point in the line are known- We may there- 
fore determine an asymptote by finding a value of for 
which r=oo or ^ = 0, and then calculating the coordinates 
(§ 138) of T , the extremity of the corresponding polar sub- 
tangent, viz. ( %7z+d, —j , remembering that the asymptote 
and radius vector must be parallel. 



154 



INFINITESIMAL CALCULUS. 



[Ch. XXXI. 



1. r = 



ad 



(Fig. 91), or u = 



Examples. 
1 1 



dd 
whence —=—ad 2 and 
du 



ad a 

r = oo or u = when 0=1. Hence the asymptote passes through 
the point (Jtt + 1, —a) or (1— J*, a) and is parallel to the line 
= 1. 





Fig. 91. Fig. 92. 

2. Find the asymptotes of the curve (r — a)d 2 = r (Fig. 92). 

Ans. Lines through (^±1, ±\a) parallel to 0= ±1. 

3. Find the asymptote of the reciprocal spiral rd = a (Fig. 85). 

Ans. A line through (%n, a) parallel to the initial line. 

4. Show that the initial line is an asymptote to the lituus 
r*d = a 2 (Fig. 86). 

5. Find the asymptotes of the curve r sin 40 = a (Fig. 93). 

Ans. Four pairs of parallel lines, each pair \a apart. 
(The numbers in figures indicate the order in which the branches 
are formed as increases from to 2n.) 

6. Find the asymptotes of the curve r 2 sin 40 = a 2 (Fig. 94). 

Ans. Four lines passing through the origin. 

7. Find the asymptotes of the curve r cos 20 = 2a. 

Ans. Four lines parallel respectively to = \n, d = \n, = |?r, 
= j7r, and passing through the points (|7r, — a), (|tt, a), 



143, 144.] 



POLAR COORDINATES. 



155 



8. Shovi that the rectangular equation of an asymptote of the 
curve r~ l =f(6) is 

J'(a){x sin cc — y cos a) + 1=0, 
where « is one of the roots of the equation /(0) = O. 



VJl 15/ 






Fig. 93. 






Fig. 94. 



A. 




143. In the curve Fig. 91, r 



Asymptotic Circles. 

ad 



a 



1-0 1__ 

6 



a if = ± 00 , 



and hence the circle of radius a is called an asymptotic 
circle. The curve approaches the circle from the outside 
when increases from the value 1, and from the inside when 
6 decreases from the value 0. Similarly r = a is an asymptotic 
circle of the curve r(6 2 —l) = ad 2 (Fig. 92). 



Points of Inflexion. 

144. Whenever the extremity of the radius vector passes 

through a point of inflexion, the perpendicular on the tangent 

is a maximum or a minimum, and hence dp/dr changes sign. 

1 /du\ 2 

Differentiating -o = =u< 2 + (-Tn) (§ 138) we have 



2 . rt , 2 du d 2 u n T / , d 2 u\ 



P 



dO 2 



156 



INFINITESIMAL CALCULUS. 



[Ch. XXXI. 



Also r=l/u, dr=—du/u 2 « 

dp o q / , d 2 u\ 



> 



d^u 
Hence at a point of inflexion u + -Tn- 2 changes sign. 



Examples. 

1. Find the points of inflexion on the curve (r — a)0 2 = r or 

au = l- 6- 2 (Fig. 92). Ans. ( + VS } fa). 

2. Find the point of inflexion on the curve r(l — 6) = ad (Fig. 91). 

Ans. is a root of the equation 3 -0 2 -2 = O, .*. (§50) 
= 1.696 rdn. = 97°% and .'. r= -2*437a. 

3. Find the points of inflexion on the lituus r 2 = a 2 (Fig. 86). 

Ans. (4, ±aV2) 

4. In the lemniscate r 2 = a 2 cos20 (Fig. 27) show that dp/dr 
= 3 cos 20, and hence that the origin is a point of inflexion on each 
branch. 

5. Show that a curve is concave or convex to the origin accord- 
ing as u + d 2 u/dd 2 is + or — . 

Multiple Points. 

145. The equation of a curve being r = /(#), the direction of 
the curve at the origin is determined by the values of d, 





Fig. 95. 



Fig. 96. 



which satisfy the equation /(0) = O. If this equation have 
two or more roots there will be a multiple point at the origin. 



145, 146.] 



POLAR COORDINATES. 



15' 



Examples, 

1. In the lemniscate r 2 = a 2 cos 26 (Fig. 27) the equation 
cos 20 = gives 6 = ±\n for the directions of the tangents at the 
origin. 

2. Find the tangents to the curve r = asin40 (Fig. 95) at the 
origin. Arts. 6 = 0, \n, \n, \n. 

These lines are also tangents to the curve r 2 = a 2 sin 40 (Fig. 96) 
at the origin. 

3. Find the tangents to the curve r = a sin 36 at the origin. 

Arts. 6 = 0, \n, §7r. 

4. Show that the curve (r — a)6 2 = r (Fig. 92) has a cusp at the 
origin. 



Curvature. 

146. Let PD, QD be consecutive normals (see §84), and 
let the angle PDQ = J(f>. We shall first show that DP-DQ 
is an infinitesimal of at least the 
second order, PQ or A$ being of 
the first. 

Draw QF perpendicular to DP. 
Then 

DP- DQ = FP-DQ(l- cos Aj>). 

But 1 — cos dcf) is of the second 
order; so is FP, since it = chord 
PQXcos FPQ, and each factor is 
infinitesimal. Hence DP-DQ is of 
at least the second order. 

Let PD = n, then QD may be 
written n + L 

Let OP = r, OQ^r + Jr, OT = p, OT' = p + Jp. Then in the 
triangle OPD 

OD 2 = P0 2 + PD 2 -2PO . PD cos OPD 

= r 2 + n 2 —2rn sin (Jj = r 2 + n 2 —2pn. 




158 INFINITESIMAL CALCULUS. [Ch. XXXI. 

Hence in the triangle OQD 

OD 2 =(r + Jr) 2 + (n + I) 2 -2(p + Jp)(n + I). 
Equating and simplifying, 

= 2r At— 2nJp + I ly .\ £n = £(r Jr/Jp). 
But £n = R, the radius of curvature PC, § 84. 

■'■ # = *■?-• (1) 

dp - 

Hence also (§§ 138, 144) 

/du\ % 



R= ' 



P+Q] 



"'*■(» +^) * 8 ("+sp) 

Examples. 

1. Find the radius of curvature at any point of r m cos md = a m 
or pr m ~ 1 = a m . 

-4ns. i2 = —-7 — — = — -z — . 

(m — l)a m (m — \)p 

If r m = a m cos m0, # = — : = — -. 

(m + l)r m - 1 (m + l)p 

2. The equation r 2 = p 2 + a 2 represents an involute of a circle, 
find R. 

3. In the logarithmic spiral r = a d , p = r sin '</>, and $ is con- 
stant, hence R = r/sm ^ = the polar normal. 

4. Show that the evolute of the logarithmic spiral is an equal 
logarithmic spiral. 

[OC is a radius vector and PC a tangent to the evolute, and in 
this case the angle OCP=(/>, a constant.] 



147.] POLAR COORDINATES. 159 

5. Prove that in any curve 

„ [-(5)7 

R = 



o ~ /dr\ 2 d 2 r* 

[ We have w = 1 /r, du=~ —dr/r 2 , d 2 u= — (r 2 d V — 2r dr 2 ) /r 4 , to 
substitute in (2) ] 

6. In the spiral r = ad (Fig. 84), #= (a 2 + r 2 )§/(2a 2 + r 2 ). 

7. In the spiral r0 = a (Fig. 85), R = r(a 2 + r 2 )%/a 3 . 

8. If a curve touch the initial line at the origin, prove that 
R = the limit of \r/Q at that point; and hence show that the radius 
of curvature of the curve of Fig. 91 at the origin is half the radius 
of the circle in the figure. 

9. Find R for the curve r = a sin nd at the origin. Arts. \na, 

10. Prove that the intercept of the circle of curvature on the 
radius vector of any curve = 2p dr/dp. 

In the curves r m cos md = a m and r m = a m cos md show that these 
chords= — 2r/(m — 1) and 2r/(ra-fl), respectively. 

Areas, etc. 

147. Let AOP=d, POQ = dd, OP = r y OQ = r + Jr. (1) The 
area-increment POQ lies between the circular sectors 
POD, EOQ, whose areas are \r 2 dd, 
%(r + Jr) 2 dd. .'. 8LTe&POD = ir 2 dd + L 

Hence the area between the curve 
and two radii vectores is 



J a 



r 2 dd. 




(2) The area bounded by two radii 
vectores and two given curves r 1 = / 1 (^) and r 2 = } 2 (0) is 



(rf-rftdd or %\\r 2 2 +n 2 ) dd 

a J a 



according as the curves lie on the same side or on opposite 
sides of the origin. 



160 



INFINITESIMAL CALCULUS. 



[Ch. XXXI 



(3) The length s= lds = Vr 2 dd 2 +dr 2 , 



taken between assigned limits. 

(4) The area of the surface formed by the revolution of 
the curve about the initial line is (§ 125 (4) ) 



2tt 



r sin d ds. 



Examples. 
1. The cardioidr = acos 2 i0 £Fig. 89). 



(1) The area = J a 



cos \Q dd = %na 2 . 



(2) The length-element ds = \ // r 2 dd 2 -\-dr 2 = a cos \6 dd, which 
does not change sign while 6 increases from — n to n 9 hence the 
whole length of the curve is 



a 



cos \d dd = 4a* 
(3) The surface of revolution about the initial line 



= 2tt 



r sin 6 ds = 2n 



= 27ia 3 



a cos 2 J# . sin 6 . a cos J# dd 



cos 4 i# sin \ 
Jo 



\0d0 = \na % . 



(The volume = \na\ § 178, Ex. 6.) 

2. The spiral of Archimedes r = ad (Fig. 84). 

(1) Let it be required to find the area included between the 
nth and (n + l)th spires. On the former r = a[2(n — l)n + ff] t and 
on the latter r = a[2nn + 0], hence the area between them 

(2n7t + d) 2 -(2(n-l)x + d) 2 dd = 87t 5 a 2 n, 



= W 



o L. 



and is .*. proportional to n. 



* A change in the sign of the length-element indicates a cusp, 
which occurs in this case when 0=7?. As increases the area-element 
\r 2 dd can change sign only with r 2 , i.e., when r becomes imaginary. 
Hence if we had integrated between the limits and 2n we should 
have obtained for the length, whereas the area would have been the 
same as above. 



147.] POLAR COORDINATES. 161 

(2) Show that the area of the first spire {0 varying from 
to2;r) =87rV/6. 

(3) The length of the curve from the origin to r = r x is 



1 

a 



Wa 2 + r 2 dr (see § 112, Ex. 2). 
o 



This is easily shown to be the same as the length of the parabola 
y 2 = 2ax from the vertex to y = r l . 

3. The lemniscate r 2 = a 2 cos 2d (Fig. 27). 

(1) The area = a 2 . 

(2) Show that rds = a 2 dd. 

(3) The surface of revolution about the axis = 27ra 2 (2 — V2). 

(4) The surface of revolution about a tangent at the centre 
= 47ra 2 . 

[This tangent being taken as initial line, the equation becomes 
r 2 = a 2 sin 26.] 

f r dv 

4. Prove that the length of any curve = ' . , and that the 

J v r 2 —p z 



_ i 



p ds = 2 



pr dr 



area = i 

5. To find the length and area of the logarithmic spiral r = a d 
(Fig. -87). 

(1) Let </> be the constant angle between the radius vector 
and the tangent. Then ds = dr /cos 0, whence 



.<? = 



' r 2 dr 



i 



COS (p COS </>' 



where r x and r 2 are the radii vectores of the extremities of the arc. 
(2) For the area, ip ds = %r sin <p dr /cos </>. 

;. area = itan^ r dr = \(r 2 2 —r± 2 ) tan $. 

6. The length of the spiral r = e~ 6 from = to # = oo is V2*. 

7. In the curve r 2 = a 2 sin4# (Fig. 96) show that the area of 
each loop = \a 2 . 



162 INFINITESIMAL CALCULUS. [Ch XXXI. 

8. In the curve r = a sin 40 (Fig. 95) show that the area of each 
loop = r^a'^i that of the circumscribed circular sector (centre 
the origin). 

- 1 - i i . 

9. Prove that the length of the curve r n =a n cos — 6 is 

n 

n(n — 2) ... 

. 2aa, 



(n-l)(n-3) . . . 



where « is 1 or \% according as n is even or odd. 

10. In the spiral r6 = a (Fig. 85) show that the area bounded 
by two radii vectores and the curve is ia(r 2 —r 1 ). 

11. The polar equation of the cissoid (Fig. 41) is r cos 6 = a sin 2 0, 
that of its asymptote is r cos = a, that of the circle of diameter 
a is r = a cos 6; show that the area between the cissoid and its 
asymptote = f^a 2 , and that the area between the cissoid and the 
circle = (^7r — l)a 2 . 

12. Find the area of a sector of the rectangular hyperbola 
r 2 cos 26 = a 2 (Fig. 98) between = and 0= a. 

Arts. \a 2 log tan ( \n + a). 

13. Find the area of a sector of any hyperbola between 6 = 
and 6 = a, the centre being the origin and the transverse axis the 

initial line. , 71 /fr + atana:\ 

Ans. iab log ( r t I . 

4 ° \b — a tan a) 

14. Find the area of an elliptic sector between 6=0 and 6=a 9 
the centre being the origin and the major axis the initial line. 



Ans. iab tan -1 (— tan.al . 



15. Show that the area of the limagon r = n(l + e cos 6) is 
nn 2 (l+ie 2 ). 

16. The chord which is drawn through the origin so as to cut 
off from a given curve a segment of maximum or minimum area 
is bisected by the origin. 

For d (area) = \r 2 d6-\r 2 d6 = 0, .*. r 1 = r 2 . 

17. Find the area enclosed by the curves 

(1) r 2 = a 2 cos 2 # + 6 2 sin 2 #. Ans. in(a 2 + b 2 ). 

(2) r 2 = a 2 cos 2 #-& 2 sin 2 0. ab + (a 2 - b 2 ) tan" 1 (a/6). 

18. The area of the common parabola r cos 2 %6 = a from 6 = 
to 6 = a is a 2 (tan ^a + J tan 3 Ja). 



148.] POLAR COORDINATES. 163 

19. If the conchoid r = a sec 6 — b has a loop, show that the area 
of the loop is aV / 6 2 — a 2 + b 2 cos~ l (a/b) — 2ab cosh-^b/a). 

148. It is in general impossible to obtain the area exactly 
unless one coordinate can be expressed in terms of the 
other, or each in terms of a third variable. 

When the rectilinear equation of a curve consists of terms 
of two dimensions only, both x and y are expressible in 
terms of m, the slope of the line drawn from the origin to 
(x, y). We can sometimes obtain the area by taking m 
as the variable. 

If m = tan d, dm = sec 2 # dd, .'.%r 2 dd = \r 2 cos 2 # dm = \x 2 dm. 



.'. the area=i 



2 

j 



x 2 dm. 



The area included between two curves will be 



(x 2 2 ±xi 2 ) dm. 



Examples. 



1 . The ellipse ax 2 + bxy + cy 2 = k. 

Substituting mx for y, we have x 2 = k/(a + bm + cm 2 ). 

Hence the whole area 



-I 



00 kdm 2nk_ 

_na + bm+cm 2 ~~ V^ac-b 2 



2. (1) The folium x 3 + y* = 3axy (Fig. 28). 
Here x = Sam /(l +m 3 ), . V area of the loop 

,Q0 9a 2 m 2 dm 



(1+m 3 ) 



= %a 2 = WBAC. 



2 ~2™ ~ 3 



(2) On the asymptote x + y-ha = 0, m= — a/(l+m); hence 
the area in the second and fourth quadrants between the curve 
and the asymptote 

if 00 f a 2 9a 2 m 2 -1 2 

Adding \a 2 , the area of the triangle ODE, we have the whole 



164 INFINITESIMAL CALCULUS. [Ch. XXXI. 

area between the curve and the asymptote = fa 2 = the area of the 
loop. 

3. Find the area of the closed part of the curve a 2 ir(y — x)+x'° = 0. 

Ans. Tta 2 . 

4. Find the area of a loop of the following curves: 

(1) ay*-3ax 2 y = x'- 7 Fig. 36. Ans. IfVla 2 . 

(2) ay A -axS?=x\ Fig. 37. jha\ 

(3) x*+y 4 = ±a 2 xy. i~a 2 . 

(4) ax* ~-y 3 = axy. ^&a 2 . 



CHAPTER XXXII. 
ASSOCIATED CURVES. 

Inverse Curves. 

149. If on the radius vector r of a curve, a distance r' 
is measured from the origin so that rr' = k 2 , where k is con- 
stant, the locus of the extremity of r f is called an inverse 
of the given curve. The radius vector of the inverse curve 
is proportional to the reciprocal of that of the given curve, 
and its polar equation may be found from that of the given 
curve by substituting k 2 /r for r. Thus (see § 139) the 
inverse of the equilateral hyperbola with reference to the 
centre is a lemniscate (Fig. 98), that of a conic section with 
reference to a focus is a limagon of the form Figs. 88, 89, 
or 90, according as e is < , =, or >1, i.e., according as the 
conic is an ellipse, parabola, or hyperbola. 

Examples. 

1. Show that the inverse of a circle with reference to a point 
on the circumference is a straight line, and that with reference 
to any other point it is a circle. 

2. The angle between the radius vector and the tangent at 
any point of the inverse is the supplement of the corresponding 
angle in the given curve. 

For, if OPQ, OP'Q' (Fig. 98) are consecutive (see § 84) radii 
vectores meeting one curve in P, P', and the other in Q, Q', the 
rectangles OP . OQ, OP r . OQ' are equal, .'.a circle may be described 
through P, Q, P', Q', .'. Q'P'P + PQQ' = two right angles; hence, 
supposing P' to approach P, the tangents at corresponding points 

165 



166 INFINITESIMAL CALCULUS. [Ch. XXXII. 

P ana Q make supplementary angles with the common radius 
vector. 

Otherwise thus : r = k 2 /r', .' . log r = log k 2 — log r' f 

.'. —dr/dd= — -dr'/dd, or cot <P=— cot ^', .*. ^' = 7r — ^. 

3. Hence show that the inverse of a logarithmic spiral with 
reference to its origin is a logarithmic spiral. 




Fig. 98. 

4. Show that the curves Figs. 84 and 85, 93 and 95, and 94 
and 96 are the inverse of each other. 

Pedal Curves. 

150. The locus of the foot of the perpendicular from a 
given point on the tangent to a given curve is called the 
pedal of the curve with reference to the point. For ex- 
ample, the pedal of a parabola with reference to its focus 
is a straight line (the tangent at the vertex); the pedal 
of an ellipse or hyperbola with reference to a focus is a circle 
(the auxiliary circle). 



150.] ASSOCIATED CURVES. 167 

In Fig. 97, T and T f are consecutive points on the pedal, 
corresponding to P and Q on the given curve; and the limit 
of position of TT f produced is the tangent to the pedal. If 
the angle AOT = cf) and OT = p, then (<f>, p) are the polar coor- 
dinates of the point on the pedal corresponding to (0, r) 
on the given curve. If then we can express p in terms of r, 
and cf) in terms of 0, the polar equation of the pedal will 
be easily obtained from that of the given curve. 

Examples. 

1. The pedal of an equilateral hyperbola is a lemniscate (Fig. 98). 

For pr = a 2 (Ex. 6, § 141), and <t> = d (Ex. 2, § 141), hence sub- 
stituting in r 2 cos 26 = a 2 we have a 2 cos2<f> = p 2 , or writing 6 
and r for $ and p, r 2 = a 2 cos 26, the equation of the lemniscate. 

In a similar way it may be shown that the pedal of any curve 
of the form r m cos m6 = a m is r n cosn#=a n , where n=m/{l—m) J 
and that the pedal of r m = a m cos m6 is r n = a n cos nd, where 
n=m/(l +m). 

2. The angle between the radius vector and tangent at any 
point of the pedal = that between the radius vector and tangent 
at the corresponding point of the given curve. 

For, in Fig. 97, let OP produced meet T'Q in S. Then 0, T, T', 
S are on the circumference of a circle since the angles at I 7 , T' 
are right angles, .'. OT'T = 0ST, and the limits of these angles 
are the angles referred to in the enunciation. (In Fig. 98, OPT 
= 0TV, if PT and TV are tangents.) 

3. Prove that the pedal of a circle with reference to any point 
is a limagon of the form Figs. 88, 89, or 90, according as the point 
is inside, on, or outside the circumference. (These figures are 
the pedals with reference to of the circles with centres B and 
radii B A.) 

4. Show that the pedal of a logarithmic spiral with reference 
to its origin is also a logarithmic spiral. 

5. Find the pedal of a parabola with reference to its vertex. 
Arts, r cos 6 = a sin 2 #, the polar equation of the cissoid, Fig. 41. 

(The directrix of the parabola is the asymptote of the cissoid.) 

6. Show that the pedal of the involute of a circle is a spiral 
of Archimedes. (It will be found that tan # is proportional to 
the radius vector.) 



168 INFINITESIMAL CALCULUS. [Ch. XXXII. 

7. Find the pedal of the ellipse with reference to the centre. 

Ans. r 2 = a 2 cos 2 # + b 2 sin 2 0. 

Polar Reciprocals. 

151. The inverse of the pedal of a curve (both pedal and 
inverse being taken with reference to the same point) is 
called the polar reciprocal of the given curve. 

Examples. 

1. Show that the polar reciprocal of a circle with reference to 
any point is a conic section. 

2. Find the polar reciprocal of a parabola with reference to 
its vertex and with reference to its focus, of an ellipse with refer- 
ence to its centre and with reference to its focus. 

3. Show that the polar reciprocal of a logarithmic spiral, r = a d t 
with reference to its origin is another logarithmic spiral. 

Roulettes. 

152. When one curve rolls on another, the curve described 
by any point connected with the rolling curve is called a 
roulette. 

The simplest case is the cycloid, the properties of which 
have already been considered. Any involute of a curve 
may also be regarded as the roulette traced by a point in 
the tangent of the curve as it rolls round the curve. 

153. The property of the normal of the cycloid holds 
for all roulettes, viz., the normal to the roulette at the tracing 
point passes through the point of contact of the fixed and 
moving curves, since at each instant the point of contact 
may be regarded as an instantaneous centre of rotation. 

154. When a circle rolls on a straight line any point not 
on the circumference describes a curve called a trochoid, 
the equations of which are easily shown to be 

x = ad—b sin 0, y = a—b cos d, 

where a is the radius of the circle and b the distance of the 
tracing point from the centre (axes as in Fig. 19). 



151-155.] 



ASSOCIATED CURVES. 



169 



155. When the circle rolls on the circumference of a fixed 
circle, the curve described by a point in its circumference 
is called an epicycloid or a hypocycloid according as the circle 
rolls on the outside or inside of the fixed circle. Corre- 
sponding to these curves we have epitrochoids and hypo- 
trochoids described by points not in the circumference. 




Fig. 99. 



For the coordinates of any point P (Fig. 99) on the epi- 
cycloid we have 

x=OB=OD cos 6-PD cos (0 + 6'). 



Hence, since arc PE = bd' = AE = ad, 



'a-{-b y 



x= (a + b) cos 0— b cos ( — — ) d. 
Similarly, y=(a + b) sin 6—b sin ( — 7— ) d. 



170 * INFINITESIMAL CALCULUS. [Ch. XXXII. 

The x and y of a point on the hypocycloid may be obtained 
n a similar way (or from the epicycloid by changing the 
sign of b), and are 

x= (a— b) cos + b cos (— r~)0- 

y=(a—b) sin 0— b sin ( — — J 0. 

The equations of the epitrochoid and hypotrochoid are 
of the same form, the coefficient b in the second term being 
changed into h, where h is the distance of the tracing point 
from the centre of the rolling circle. 

Examples. 

1. Show that in any epicycloid 

ad b 6' 

ds = 2(a + b) sin-dd = 2{a + b)— sin-dd', 
26 a 2 

and hence that the length of the curve from cusp to cusp is 
8(a + b)b/a r f 

2. Show that the epicycloid is a cardioid when b = a. 

3. Show that the hypocycloid is the curve x%+y% = a$ (Fig. 18) 
when b = \a. 

4. When a circle rolls inside another circle of double its diam- 
eter, show that every point in the circumference describes a straight 
line and every other point an ellipse. 

5. The radius of curvature at any point of an epicycloid 

4(a + b)b ad 4(a + 6)6 . 6' 2(a + b) , , nn 

= — -. sin — = —7- sin — = — X chord EP 

a + 2b 2b a + 2b 2 a + 2b 

and is therefore proportional to the chord EP. 

For, if the tangent at P make an angle <j> with OX, $ = 6 + ^6' 
£LndR = ds/d<f> (§85). 

6. Show in a similar way that in the cycloid 

x = a(6 — sin0), y = a(l—cosd), Fig. 60. 
ds = PB dd, 4>-h^-¥> and hence that R = 2PB. 



156.] 



ASSOCIATED CURVES. 



171 



Envelopes. 

156. Let f(x,y f a)=0 represent the equation of a curve 
(i.e. of any plane locus, including a straight line), a being a 
quantity involved in the equation, but independent of x 
and y for its value. As a may have any value the equa- 
tion may be regarded as representing a family of curves. 
Supposing a to have a certain value in one instance, let it 
receive an increment J a. The two equations 

f(x,y,a)=0 (1), f(x,y,a + Ja)=0 (2) 

then represent two curves of the family. Their points of 
intersection approach limits of position as Aa = 0. The locus 
of these point-limits for all values of a is called the envelope 
of the family of curves. 

The quantity a is called a variable parameter. 




Fig. 100. 

Ex. y=a 2 x + a represents a family of straight lines. Consider 
the two for which a is 1 and 1+Ja respectively. The lines y = x + l, 

y=(l + Ja) 2 x + (l+Ja) intersect in the point ( — , ). 

\ 2 + A a 2 + da/ 

The limit of this when Ja = is ( — J, J). This is therefore one 

point on the envelope of the family. 



172 INFINITESIMAL CALCULUS. [Ch. XXXII. 

157. Equation of the envelope. The points of intersec- 
tion of (1) and (2), § 156, lie on the curve 

f(x, y,a + Ja)-f(x, y, a) . 

since this equation is satisfied by any simultaneous values 
of x and y which make f(x,y,a) and f{x,y,a + Aa) sepa- 
rately =0. As Ja = the limit of (3) is 

3/0, y, a) 



da 



= 0, (4) 



the differentiation being partial since only a varies. The 
point-limits of the intersections of (1) and (2) therefore 
lie on (4), and their locus, the envelope, is obtained by 
eliminating (a) from (1) and (4). 

Ex. Equation (4) for y=a 2 x + a is 2a:r + l = 0. Eliminating a, 
4:xy=— 1 The envelope is therefore a rectangular hyperbola 
(Fig. 100). 

158. Prop. The envelope touches every curve of the family. 
Let u stand for f{x, y, a), and suppose (x, y) to be a point 
common to the envelope and curve (1), § 156. For dy/dx, 
the slope of the tangent of (1), we have (§ 47), 

fx dX+ ¥y dy = °> 

a being constant. We may consider (1) to be also the 
equation of the envelope, a being a variable, viz., that 
function of x and y obtained from (4) . Hence for the envelope 

du , du du, 

o ax + —ay+— da = 0. 

ox oy ool 

But du/da = from (4). Hence dy/dx at (x,y) is the 
same for (1) and the envelope. 

Ex. In the example of §§156, 157, y=x + l and the envelope 
4x2/ = — 1 touch at the point (-—J-, i). 



157-159.] 



ASSOCIATED CURVES. 



173 



159. The given equation may contain two ol* more vari- 
able parameters, subject however to other relations con- 
necting them, whereby all except one may be eliminated 
from the given equation, 

Ex. To show that all ellipses having the same centre and 
area, and their axes in the 
same directions, touch a pair 
of hyperbolas of which the 
axes are asymptotes. 

We have to find the envelope 
of x 2 /a 2 + y 2 /P 2 = l, where a/3 
= k 2 , a constant, whence 

Substituting, the equation 
becomes x 2 /a 2 + a 2 y 2 /k 4 = l. 

Differentiating with regard 
to a, -2x 2 /a 3 + 2ay 2 /k* = 0. 

Eliminating a, xy = ±%k 2 , j? IG ^ 

the envelope (Fig. 101). 

Examples. 

1. Two sides of a right-angled triangle are given in position 
and the area is constant, find the envelope of the hypotenuse. 

Arts. A rectangular hyperbola. 

2. Particles are projected in the same vertical plane with the 
same velocity v, but at different elevations; show that their paths 
all touch the parabola 

2v 2 / v 2 \ 




X 



2_ 



of which the point of projection is the focus. 
[In other words find the envelope of 

y = x tan a — gx 2 /(2v 2 cos 2 a).] 

3. Show that the circles described on the double ordinates of 
the parabola y 2 = Aax as diameters touch the equal parabola 
y 2 = 4ia(x + a). . 

4. Find the envelope of ua 2 + va + w = 0, where u, v, w are 
functions of x and y. Arts, v 2 = 4tuw. , 



174 INFINITESIMAL CALCULUS. [Ch. XXXII. 

The result is the same as the condition that the given equation 
should have equal roots. Explain. 
5. Find the envelopes of 

ucos m d + v sin m d = w, (1) 

u sec m # - v tan m = w. (2) 

Arts. (1) u n + v n = w n ) , 2 

where n 



(2) u n — v n = w n ) 2— ra' 

Many examples may be reduced to these* by observing that a 

condition of the form ( — J + (t - ) = 1 is equivalent to the two 

relations a = a cosr d, /? = & sin?* d y while ( — ) — ( t-) = 1 is equiva- 

lent to a = a sec r d, /? = b tan r d. 

6. Find the envelope of a line which moves in such a way that 

the sum of its intercepts on the axes is constant. 

x v 
We have — +— = 1, and a +/? = k. We may substitute the value 

of /? and then differentiate, or we may proceed as follows: 

Let a = k cos 2 d, /? = /csin 2 #; the line becomes #(cos 6)~ 2 + 
y(sin d)- 2 = k, hence (Ex. 5) the envelope is x* + y$ = kb, & parabola 
touching the axes. 

7. A straight line of given length k moves with its extremities 
on two rectangular axes, find the envelope of the line. 

Arts. xi + y$ = ki, a four-cusped hypocycloid. 

8. Given in position the axes of an ellipse and that their sum 
= 2k, show that the ellipse touches the curve x$ + y$ = k$. 

(x\ m ( y \ m 
— 1 ± I— \ =1 perpendiculars are 

drawn to meet the axes in A and B, find the envelope of AB. 

(x\ n /y\ n , . m 

Arts. [ — ) ± — ) =1, where n= -. 

W \6/ m + 1 

x 2 y 2 
10. To the ellipse or hyperbola —±— = 1 pairs of tangents are 

x 2 y 2 
drawn from points in the ellipse — ^+— = 1, show that the chords 



159.] ASSOCIATED CURVES. 175 

of contact touch the ellipse 

(f) '+(£)'-'• 

11. When the tangents are drawn from points in the hyperbola 

x 2 ii 2 

~— f- = l, show that the chords of contact touch the hyperbola 

a 2 b 2 



(?)-(f)-i. 



12. The e volute of a curve may be considered to be the en- 
velope of its normals ; find in this way the evolute of an ellipse. 

cl 2 x b 2 y 
The normal at {a /?) is = a 2 — b 2 , 

a (j 

or, writing a cos 6 for a, and b sin 6 for /?, 

x . a(cos 0)- —y.b (sin 0)- 1 = a 2 — b 2 , 
the envelope of which is (Ex. 5), 

(ax)i + (by)$ = (a 2 -b 2 )i, 

which is therefore the evolute (cf. § 89). 

13. Show in a similar way that the evolute of the hyperbola is 

(ax)t-(by)$=(a 2 + b')i. 

14. Parallel rays of light are reflected from the circumference 
of a circle. Find the envelope of the reflected rays. 

Take the centre for origin and the x-axis parallel to the incident 
rays. The equation of the ray reflected from the point (a cos 0, 
a sin 0) is 

x sin 20 — y cos 20 — a sin = 0, 

whence the envelope is 

x=-{a(3 cos — cos 30), ?/ = |a(3 sin 0-sin 30), 

an epicycloid formed by a circle of radius \a on a circle of radius £a. 



CHAPTER XXXIII. 
CENTRES OF GRAVITY. 

160. In finding the coordinates x, y of the centre of 
gravity of a body, we 'suppose the body to be divided into 
parts of weights W\, w 2 - . -, and of which the centres of gravity 
are the points (xi, yi), (x 2 , y 2 ), • • • , then equate the sum 
of the moments of the weights to the moment of the sum 
of the weights if placed at the centre of gravity. Thus 
supposing gravity perpendicular to the z-axis we have 

WiXi+W 2 X^ + . . . = (wi+w 2 + . - - )$y 
-__WiXi~\-W2X 2 -\-. . ._IWX 
W1+W2 + . . . " Iw ' 

Similarly supposing the body and the axes placed so that 
gravity is perpendicular to the y-axis, we have 

- = wiyi+w 2 y2 + - • - __ 2wy 

W\ + w 2 + . . . Iw ' 

These formulae also hold when the points are not in one 
plane, there being also a third coordinate, 

~z = Iwz/Iw. 

If the parts referred to are infinitesimal the sign of inte- 
gration replaces that of summation to indicate the limit of a 
sum. 

161. The division into parts and the limits of the sum- 
mation in the following cases are the same as if we were 
about to calculate an area, volume, or length. The bodies 
are assumed to be homogeneous (of uniform density) and 

176 



160-1G4J 



CENTRES OF GRAVITY. 



177 



hence weight is proportional to volume. For such bodies 
the centre of gravity is also known as the centroid. 

162. An area. To find the e.g. of a thin plate or lamina * 
of the form ABDC, Fig. 61, we have the element of area = y dx; 
element of w T eight = w . y dx, where w = weight per unit area; 
e.g. of element at (x+i, \y), where i is infinitesimal; hence 
element of moment = wy dx . x when gravity is perpendicular 
to the x-axis, smd = wy dx . \y when gravity is perpendicular 
to the y-axis. Dividing the sum-limit of the moments by 
that of the weights (§ 160), 



xy dx 



y 2 dx 



x = 



V = h 



II 



dx 



y dx 



when w (which is assumed to be constant) is cancelled. 
^It will be noticed that the denominator = the area. 

163. A solid of revolution about OX. Element of vol- 
ume = xy 2 dx, of weight = w . ~y 2 dx, w being the weight per 
unit volume, element of moment = w . ~y 2 dx . x, 



. . *v — 



xy 2 dx 



= 0. 



y 2 dx 



The denominator = volume/7r. 

164. An arc. Proceeding as above we have for the e.g. 
of a material line in the form of the curve CD,t 



x ds 



yds 



x 



y 



ds 



ds 



* Results for a lamina are limits for a uniform and infinitesimal 
thickness. 

f The results are limits for a body of uniform and infinitesimal 
cross-section. 



178 



INFINITESIMAL CALCULUS. [Ch. XXXIII. 



165. A surface of revolution about OX. For a curved 
surface of this form and of infinitesimal thickness, 



x = 



xy ds 
y ds 



y = 0. 



166. An area in polar coordinates. Element of area = 

\r 2 dd (§ 147) ; its e.g. is distant §r + i from the origin;* 
hence if the initial line is taken as a>axis, 



x = 



• 

w . 1 


■r 2 dd 


| 


-r cos 6 


• 

r 3 ccs dd 


_ 2- 1 






r 3 


r 


• 


w . \r 2 dd 

a 


r 2 dd 




r 3 sin 6 dd 


V-i r 








r 2 dd 





and 



167. The subject may also be considered from the point 
of view of geometry only. Let a be an area-element or 
volume-element which is infinitesimal in every direction, 
and which contains a point (x, y, z). Then the limits of 

lax lay laz 
la ' la ' la 

are the same as the x, y,z of § 160 for homogeneous bodies, 
and are the coordinates of a point which is called the cen- 
troid (or centre of gravity) of the area or volume. Or, a 
may be taken as a mass-element, in which case the point 
is called the centre of mass (or centre of gravity) of the body. 

168. Pappus's (or Guldin's) properties of the centre of 



gravity. From § 164 we have yds = y 

J 

both sides by 2tt, 



ds, and multiplying 



* The e.g. of a triangle is assumed to be the point of intersection 
of the medians. 



165-168.] CENTRES OF GRAVITY. 179 

\2ny.ds=l ds) .2riy. (1) 



Similarly from § 163, 



ny 2 dx = 



ydx). 2ny. (2) 



These results are equivalent to the following statements, 
which are known as Pappus's or Guldin's Properties: 

(1) The surface of a solid of revolution is equal to the 
length of the revolving curve multiplied by the length of 
the path of the e.g. of the curve (i.e. of the arc), 

(2) The volume of a solid of revolution is equal to the 
revolving area multiplied by the length of the path of the 
e.g. of this area. 

N.B. The axis of revolution may touch but not cut the 
curve. 

Examples. 

1. The parabolic area OAB, Fig. 74. Arts. x = %x iy y = iy 1 . 
Of the solid of revolution round OX, x = fa: 1 . 

2. The quadrant of an ellipse. Ans. £ = — , y~= <r- 

3. Half of a prolate spheroid. Ans. x = %a. 

4. The circle x 2 + y 2 = 2ax between x = and x=--h revolves 
about the axis of x, find the e.g. of the volume of the spherical 
segment thus formed. _ _ /8a — 3h\ h 

Ans x ~\z^=h)l' 

For a hemisphere this = fa. 

Show that for the surface of the segment x = %h. 

5 A circular arc. 

Ans. Distance from centre of circle = chord X radius/arc. 
For a quadrant this = 2V / 2a/7r, and =2aA for a semicircular 

CxJL \J» 

6. A circular sector. 

Ans. Distance from centre = f chord X radius/arc. 

For a quadrant this = 4v / 2a/37r and =4a/37r for a semicircle. 



180 INFINITESIMAL CALCULUS. [Ch. XXXIII. 

7. A circular segment. . 

Arts. Distance from centre = chord 3 /(12X area of segment). 
8 Surface and volume of a right circular cone. 

Arts. Distance from vertex = (1) § axis, (2) f axis. 

9. The area between the curve y = smx (Fig. 65) and the axis 
of x, from x = to x = n. Arts, x = \n } y = \n. 

10. The cycloid, Fig. 19. Ans. Distance from base = |a. 

11. A quadrant of the curve x% + yl = a\ (Fig. 18). 

Ans. x = 256a/3157: = y. 
Of the arc, x = %a = y. 

12. A quadrant of the whole area of the curve a 2 y 2 = x 2 (a 2 — x 2 ) y 
(Fig. 69). Ans. x = ^na, y = \a. 

13. The area between the curve y 2 (a 2 —x 2 ) = a x and the asymp- 
tote x = a. Ans. x = 2a/n y y = 0. 

14. Find by Pappus's Properties the surface and volume of a 
torus (or anchor ring), formed by the revolution of a circle of 
radius a when the centre describes a circle of radius b, b> a. 

Ans. 4:7t 2 ab, 2n 2 a 2 b. 

15. Find by Pappus's Properties the e.g. of the arc of a semi- 
circle and that of the area of a semicircle. 



CHAPTER XXXIV. 
MOMENTS OF INERTIA. 

169. The Moment of Inertia is a quantity which is often 
required in connection with the motion of a body about an 
axis. The following is an illustration. 

170. Kinetic energy of rotation. Let it be required 
to find the kinetic energy which a body possesses on account 
of its rotation about an axis. 

Let the perpendicular distance of a particle of mass mi 
from the axis be r x and let aj = the angular velocity of the 
body about the axis. Then the kinetic energy of the par- 
ticle 

= J (mass) X (linear velocity) 2 ^ \m± (a>ri) 2 = %co 2 miri 2 , 

and the whole kinetic energy of the body, 

= ^(rai7Y* + ra 2 r2 2 + . . . ) = i^ 2 /, 

where / = mir^ 2 + m 2 r 2 2 + . . . 

The quantity / is called the moment of inertia of the 
body about, or with reference to, the axis; hence the fol- 
lowing definition: 

The Moment 0} Inertia of a body about an axis is the sum 
of the products obtained by multiplying the mass of each 
particle of the body by the square of its distance from the 
axis. 

181 



182 INFINITESIMAL CALCULUS. [Ch. XXXIV. 

Since the particles of a body are infinitesimal portions 
of the body, the moment of inertia is obtained as follows: 
Imagine the body to be divided into parts which are infini- 
tesimal in every direction, and find the limit of the sum 
of the products of the mass of each part by the square of 
the distance of some point in it from the given axis. 

Since both factors of the product mr 2 are essentially + , 
the moment of inertia is always + , and the moment of 
inertia of a body about any axis is always equal to the arith- 
metical sum of the moments of inertia of its parts about 
that axis. 

171. Prop. The m.i. of a body about any axis = them.i. 
about a parallel axis through the centre of gravity + MH 2 , 

where M is the mass of the body and 
h is the distance between the parallel 
axes. 

Take at a point P in the body a 
particle of mass m\. Let a plane 
through P perpendicular to the axes 
in question meet the one through 
the centre of gravity G in G' and the 
parallel one in H' , then G'H' = h. 
Draw PK perpendicular to G'E f and let G'K = x 1 . Then 

s 1 2 =r 1 2 +h 2 -2hx 1 (Euc. II. 13), 

.*. miSi 2 = m 1 ri 2 +mih 2 —2hmiXi. 

Similarly for particles ra 2 , m 3 , etc. 

.'.m 1 s 1 2 + ra 2 s 2 2 + . . . = (m 1 r 1 2 + m 2 r 2 2 + . . .) + h 2 (m 1 +m 2 + . . .) 

— 2h(miXi+m 2 x 2 + . . . ). 

The left-hand side = the m.i. about the axis through H; 
and of the three terms on the right, the first = the m.i. about 
the parallel axis through the centre of gravity, the second = 




171-174.} 



MOMENTS OF INERTIA. 



183 



Mh 2 , and the third = (§ 160), since the centre of gravity 
is in the line from which xi, x 2 , . . .,are measured. 

172. The proposition just proved is true for all bodies, 
but the following applies only to laminae. 

Prop. Let X'X, Y'Y be two lines in the plane of a lamina 
and meeting at right angles in 0, and let Z'Z be a line through 
perpendicular to the plane. 
Let 7i = the m.i. of the lamina 
about X'X, 7 2 = that about 
Y'Y, /-that about Z'Z) then 



iWi+/ 2 . 

For ri 2 = Xi 2 + yi 2 , 
.'. miri 2 = miXi 2 + m,iyi 2 . 




Fig. 103. 



The proposition is therefore 
true for a particle at P, and hence it is true for all the 
particles of the lamina. 

173. When the m.i. is put into the form Mk 2 (M being 
the mass), k is called the radius of gyration; hence the radius 
of gyration of a body with reference to an axis is the dis- 
tance from the axis of a point at which a particle having 
the same mass as that of the body may be placed so that 
its m.i. may be the same as that of the body. 

If k = the radius of gyration with reference to an axis 
passing through the centre of gravity, and ki that about a 
parallel axis at a distance h, we have ki 2 = k 2 + h 2 , since 
(§171) 

Mk 1 2 =Mk 2 +Mh 2 . 



174. In the following examples the density, i.e., the mass 
per unit volume, is represented by fi, and the bodies are 
assumed to be homogeneous, i.e., of uniform density, unless 
the contrary is specified. 



184 



INFINITESIMAL CALCULUS. [Ch. XXXIV. 



Ex. 1. To find the m.i. of a rectangular lamina whose sides are 

a, a, b, b, about an axis bisecting the 
sides a, a, Fig. 104. 

Divide the rectangle into parallel 
strips of length b and width dx, and 
measure x from the axis. The elements 
are as follows : area = b dx, volume = 
t . b dx, where t = the thickness of the 
lamina, mass = fi.tb dx, m.i. = ptb dx . x 2 , 
since every particle is at a distance 
x + i from the axis, i being infinitesimal. Integrating between 
and \a and doubling we have for the m.i. of the whole rectangle 



o x 



-a- 



Fig. 104. 



|*2 a 2 

2 irtbx 2 dx = Tzfitba? = (v-tab)-— 
Jo 12 



The quantity in parentheses is the whole mass ( = fx X volume), 



a' 



a 



.'. the m.i. =m-— , and hence the radius of gyration = ~7Yo' Simi- 

. . . . . b 2 

larly the m i. about the axis bisecting the sides b, b is m — . 

1.JU 

2. The m.i. of the rectangle about a normal axis through the 

a 2 + b 2 
intersection of the two axes of Ex. 1 is § (172) m — — — 

The same formula is true for any parallelogram (of which a and 
b are adjacent sides) about an axis drawn as in this case through 
the intersection of the diagonals at right angles to the plane. 

3. The m.i. of the rectangle about 



a side b is (§ 171), 



a' 

m — +m . 
12 \2 



&- 



a' 



m 



4. The m.i. of the rectangle about a 
normal axis through one angle = 




a- 



i-b' 



m- 



Fig. 105. 



5. Any triangular lamina (Fig. 105) about one side BC. Let 
BC = a, the perpendicular OA = h. Then DE:BC::FA:OA, 



174.] MOMENTS OF INERTIA. 185 

.*. DE/a=(h-x)/h, .\ DE=(h-x)a/h. 



. . m.i. 



I"* a h 2 

n(h — x)— . t . dx . x 2 = m—. 
oh o 



6. A circular lamina of radius r about a normal axis through 
the centre. 

Consider the annulus between the concentric circles of radii 
x and x + dx. The elements are : area = 2nx . dx, mass = trt . 2nx dx, 
m.i. = pi . 27r# <£c . x 2 , since every particle of the annulus is at 
a distance x+i from the axis, i being infinitesimal, 



,\ whole m.i = 



r r 2 

2/intx 3 dx = hpxtr* = m~- 
o 2 



7. A circular lamina about a diameter. 

Let the required m.i. =/. The sum of the moments of inertia 
about two diameters at right angles to each other = 27; it also 



2 2 



(by § 172 and Ex. 6)=ra- ,\ l = m-~. 

A 4 

8. A circular lamina about a normal axis through the centre 
when the density is supposed to vary inversely as the distance 
from the centre. 

Let n = k/x, where k is a constant. Then m.i. 



-m ■ 



— ) . 2ntx z dx = %7ztkr 3 . 



But the mass m =\ { — ) . 2nx dx . t = 2xktr. 



-m ■ 



r 2 



.". m.i. =ra— . 
o 



x 2 y 2 



9. An ellipse — +— = 1 about its minor axis. 
a 2 b 2 

The m.i. =4 



a 

H . y dx .t . x 2 . Substitute y from the equation of 
o 



a 2 



the curve and let x=*a sin 6. The result is m-. 

4 



186 INFINITESIMAL CALCULUS. [Ch. XXXIV. 

b 2 
Similarly about the major axis the m.i. is m-r-. 

10. A sphere about a diameter. 

Take the diameter as x-axis and the centre as origin. 
Consider the sphere to be made up of laminae perpendicular to 
the axis, and of thickness dx Then m.i. (see Ex. 6) 

r y 2 

= 2 Jul . xy 2 . dx . ~2 and y 2 = r 2 —x 2 , whence the m.i. =m|r 2 . 

11. A right circular cylinder of radius r about its geometrical 
axis. 

The cylinder may be considered as made up of circular laminae 

r 2 
perpendicular to the axis, hence (Ex. 6) the m.i. =ra— . 

Similarly for a cube, a right prism, etc., about an edge or any 
parallel axis. 

12. A right circular cylinder of radius r and length I about 
an axis bisecting at right angles the geometrical axis. 

As before, suppose the cylinder to be made up of circular 

laminae. The mass of the lamina at a distance x from the axis 

= fi . nr 2 dx, and its m.i. about a diameter in its own plane and 

r 2 
parallel to the given axis = mass X— (Ex. 7), .*. its m.i. about 

(r 2 X 

the given axis = mass! — +x 2 J (§ 171); 



.\ whole m.i. = 2 



ft 1 (r 2 \ (r 2 l 2 \ 



175. As in Ch. XXXIII, the subject may be considered 
from the point of view of geometry alone. If a is an area 
(or volume) element which is infinitesimal in every direc- 
tion, and r the perpendicular distance of some point in it 
from a straight line, the limit of lar 2 is called the moment 
of inertia of the area (or volume) with reference to the straight 
line. The results are the same as those calculated for homo- 
geneous bodies, area (or volume) taking the place of mass. 



175.] MOMENTS OF INERTIA. 187 

Examples. 

Find the moment of inertia of 

1. A triangle about (1) an axis through the centre of gravity 

parallel to the base, (2) a parallel axis through the vertex. 

h 2 h 2 

Arts. (1) w— , (2) m-. 

a 2 b 2 



2. A rectangle (a by 6) about a diagonal. Ans. m 

6(a z +cr) 

3. An isosceles triangle about a normal axis through the middle 

point of the base. . 4 alt. 2 + base 2 

Ans. m — . 

24 

4. An isosceles triangle about a normal axis through the vertex. 

12 alt. 2 + base 2 

Ans. m- . 

24 

5. A circular annulus of radii R, r about a normal axis through 

the centre. A R 2 +T 2 

'Ans. m . 

2 

R 2 +T 2 

6. A circular annulus about a diameter. Ans. m . 

4 

7. A circle about a tangent. Ans. m^r 2 . 

8. A circular arc of length s, radius r, and chord c, about an 
axis through its middle point perpendicular to its plane. 

Ans. m . 2r 2 ( 1 ). 

9. A circle about a normal axis through a point in the cir- 
cumference. Ans. mlr 2 . 

v 2 

10. A parabolic area, Fig. 74, about the z-axis. Ans. m— 

5 

11. The same about the y-axis. Ans. nijx 2 . 

12. A spherical shell of infinitesimal thickness about a diameter. 

Ans. m^r 2 . 

13. A right circular cone of radius r and altitude h about its 
geometrical axis. Ans. m-f^r 2 . 

14. The same about an axis through the vertex perpendicular 

to the geometrical axis. Ans. 12h 2 +3r 2 

m 

20 



188 INFINITESIMAL CALCULUS. [Ch. XXXIV. 

15. The same about an axis through the centre of gravity per- 
pendicular to the geometrical axis. Sh 2 + 12r 

Arts, m — — . 

16. An oblate spheroid about its geometrical axis. 

Arts. m%a 2 
17 Any area ABDC (Fig. 61) about the z-axis. 



m 

Ans. — 

o 



ft 



dx 



> 



dx 



18. The hypocycloid xs+y* = a? (Fig. 18) about the x-axis. 

Ans. m-Q^a 2 



CHAPTER XXXV. 



SUCCESSIVE INTEGRATION. 



176. Successive integration. Let T be a function of u, 
v, and Wj and suppose the following operation to be per- 
formed: T dw is integrated between the limits w\ and w 2 , 
u and v being treated as constants; the result is mul- 
tiplied by dv and integrated between v\ and v 2 , u being 
treated as a constant; the result is multiplied by du and 
integrated between u\ and u 2 . The whole operation is 
indicated by the notation 

T du dv dw. 

U X J V\J Wi 

The limits of w may be functions of u and v, the limits 
of v may be functions of u, but the limits of u are constants. 
Instead of three variables there may be two or four, or more. 



! 



Ex. 1. 
2. 



X 



1, 

3 
Oj 



x 2 y dx dy = 



9 



X 



x 2 y 







dy) dx = 



C2 



%x 4 dx = 3tV. 



3. 

4. 
5. 



1. 
•2 

Ji- 
Jo. 



fz+2/ r3 rx 

(x — y)dx dy dz=\ (x 2 —y 2 )dx dy 
J0J0 

= [ %x*dx = 13h 
Jo 



6x 2 y dx dy = S5 



•2x 

. 

Cad 



Cx+y 



dx dy dz = 9 J. 







r dd dr = in 3 a 2 . 



189 



190 



INFINITESIMAL, CALCULUS. [Ch. XXXV. 



6. 



C2t: 
, 



2 
J 



2a cos 6 



r 3 sin d cos <i<:/> dd dr = $7ia i . ' 



Applications of Successive Integration. 

177. Plane area. Rectangular coordinates. Let P be the 

point (x, y), and let PR, PS be dx, dy, infinitesimal incre- 
ments of x and y. The 
rectangle PQ = dx dy may be 
taken as an element of area. 
To illustrate the method of 
finding the limit of the sum 
of such elements by succes- 
sive integration, let it be re- 
quired to find the area of the 
figure KM, which is bounded 
by the lines x = a, x = b, and 
the two curves KL or y = J(x) 
and NM or y = F{x). 




Fig. 106. 



(1) ( dy)dx=EG.dx, (2) 

V CE ' 



EG . dx=KM. 



a 



The first result is the sum of such rectangles as PQ, which 
make up the rectangle EH, which is equivalent (§ 15) to 
the strip EIJG of the given figure, the second is the sum of 
such strips for the whole area. The final result is the limit 
of the sum of such rectangles as PQ when both dx and dy = 0. 
The whole operation is indicated by the " double integral" 



'2/2 



dx dy or 



Vi 



a 



F(x) 



fix) 



dx dy, 



yi and y 2 being the y's of the curves y = f(x), y = F(x). 

More generally, let u be a function of x or y, or of both x 
and y. The limit of the sum of such products as u dx dy 
taken for all parts of the area KM is 



CV2 



u dx dy. 



J a J 2/1 



177, 178.] 



SUCCESSIVE INTEGRATION. 



191 



Ex. 1. Find £1 y dx dy for a quadrant of the circle x 2 + y 2 = a' 



We are to obtain 



fa 



CVi 



J 



y dx dy, where y x is the y of x 2 + y 2 = a 2 . 



o J 



y\ 



y dx dy = 



( y dy) dx = 
\l ' 



hji 2 dx = 



i(a 2 —x 2 )dx = ^a 3 . 



o 



This is the moment of the area with reference to the x-axis. 
2. Find £1 y 2 dx dy for the same area. 



J 



V a 2-x 2 



o 



y 2 dx dy = TG7za i . 



This is the second moment, or moment of inertia, of the area 
with reference to the x-axis. 

3. Find £1 xy dx dy for the same area. Arts. |a 4 . 
This is the product of inertia of the area with reference to the 

axes. 

4. Show that the volume of a solid of revolution about the 
x-axis is 



2tt 



y dx dy. 



and deduce the formula of § 95. 

178. Plane area. Polar coordinates. Let P be the 

point (0, r), and let POR = dd, PS = dr. With centre 
describe the arcs PR, SQ. Then 
the element of area PQ 

= ±(r + dr) 2 dd-ir 2 dd 

= rdddr + %dr 2 dd, 

the last term being a higher 
infinitesimal. 

.'. £1 PQ=£Irdddr. 

Let KLMN be a figure bounded 
by the lines 6 = a, 5 = /?, and the £ 
curves KL or r = }(0) andiVMcr Fia, 107. 




192 



INFINITESIMAL CALCULUS. 



[Ch. XXXV. 



r = F(d). Proceeding as in § 177 we find the area 



'r 2 



r dO dr, or 



r/3 



a J 



n 



rF(e) 



aj 



Kd) 



r dd dr, 



r\ and r 2 being the r's of the given curves. 

The first integral %(r 2 2 —r 1 2 )dd = EGJF, Which is equivalent 
to the sectorial strip EGHI of the given figure; the second 
is the sum of such strips for the whole area. The final 
result is the limit of the sum of such figures as PQ when 
both dd and dr = 0. 

Observe that the element r dd dr=PR . PS as in Fig. 106. 

Ex. 1. The area (Fig. 108) between the circles r = 2acos#, 
r = 2b cos 0, (b>a)j is 



2 




26 cos 6 



rdddr = 7t(b 2 -a 2 ) 



2a cos d 



2. Find the area (Fig. 109) bounded by tl\e curves d = r 3 i-r i 
d = r z — r r = L 





Fig. 108 



Fig. 109. 



[Integrate first with regard to 0.] Arts. § . 

3. The moment of inertia of the circle r = 2acos# with refer- 
ence to a normal axis through the polar origin 



= 9 



' 2 T2a cos 6 

Jo 



r 3 d0 dr = %7ia 4 . 



4. Find the moment of inertia of the lemniscate r 2 = a 2 cos2# 
(Fig. 27) with reference to a normal axis through the origin. 

Ans. jxa*. 



179.] SUCCESSIVE INTEGRATION. 193 

5. The potential of a particle of mass m at distance R being 
m/R, show that the potential of a lamina (thickness t, density /*) 
at a point B on the perpendicular to the lamina through the polar 
origin (OB = c) is 

'C/trdOdr 

J J ^/c 2 +r 2 ' 

(1) Find the potential at B if the lamina is the circle r = a. 

Ans. 2nnt(Vc 2 + a 2 — c). 

(2) Find the potential of a circular lamina at a point in the 
circumference. Ans. \airt. 

6. If a curve revolve about the initial line show that the volume 



-*■/. 



r 2 sin 6 dd dr. 



Find this volume for a complete revolution of the cardioid 
r = a cos 2 ^#, Fig. 89. Ans. \na*. 

7. Find the moment of inertia of an anchor-ring (see Ex. 14, 
p. 180) about its axis. 

Take as origin the centre of the circle (radius a) to be revolved, 
and the perpendicular (length b) on the axis of revolution as 



initial line. The m.i = 2 



o, 



a 

p. ,2n(b — r cos 0) .rdddr t (b — r cos d) 2 


Ans. m(|a 2 + 6 2 ). 



179. Volume of a solid. Rectangular coordinates. Let P 

be the point (x,y,z) and let PR, PS, PT be dx, dy, dz, infinitesi- 
mal increments of x, y, and z. The parallelepiped PQ = dx dy dz 
is taken as the element of volume, the limit of the sum of 
such elements being obtained by successive integration. 
Thus to find the volume of the solid, Fig. 110, bounded by 
the coordinate planes and a surface whose equation is given: 

([ lv \ 

(1) ( dz) dx dy=IU. dx dy, 

\ ' 

(2) ( IV .dy)dx=VDG.dx, 

rOA 

(3) VDG .dx = COBA, 
Jo 

the required volume. 



194 



INFINITESIMAL CALCULUS. 



[Ch. XXXV. 



The first result is equivalent to the column standing on 
IK( = dx dy), the second to the slice between VDG and WEF 
and therefore of thickness dx, the third is the sum of such 
slices. 




Fig 110. 
The whole operation is indicated by the " triple integral" 



a f 2/1 



J 0J J 



'21 



dx dy dz, 



where Z\ is the z of the given surface, y\ is the y of the curve 
AGB in which the surface is cut by the plane z = 0, and a = OA. 



Ex. 1. To find the volume of the ellipsoid — ++— = 1. 



IV- 



V 



o. 



2/1 

J 



a 2 ' b 2 c 2 
X s y 2 z 2 



z x ^ X' ■ y z 

dx dy dz, where z x is the z of — 2 + T2+1 = 1, and 



o 



2/iistheT/of — 2 +— = 1 ; i.e 



6 c 

2/,=-(a 2 -z 2 )* and z^-riyS-y 2 )*. 



179.] 



SUCCESSIVE INTEGRATION. 



195 



Since 



•zi 



dz = z 



o 



• • 8 v ~ 



a 



J 



Vi 



#! dx dy = 



o 



a \V\\ 



\T(yi 2 -y 2 )My]dx 
, o Jo La J 






a izbc 



o 



4a : 



(a 2 — x 2 )dx = 



nabc 



6 



-7T 



a6c. 



«i cfc cfa/ is equivalent to a column, - 2/i 2 dx to a slice, and is 

46 6 

the sum of the slices for an octant. 

2. Find £I^(y 2 + z 2 )dx dy dz for an ellipsoid, i.e., the moment 

of inertia with reference to the #-axis. 



8 



o, 



'2/1 







Zl 



[i{y 2 + z 2 )dx dy dz = &fJL 



o 



a 



'2/1 



(2/ 2 £i +isi 8 )dz dy 



-£»■+•■> 



y A 4 dx = i%7zabcii(b 2 + c 2 )=m 



b 2 + c< 



o 



3. Find the volume of the hyperbolic paraboloid az = x 2 — y 2 in 
the first octant, # varying from to h. 

x 2 —y 2 

1 a 4 



rfc 



Cx 



0, 



a 







cfo dy dz = . 

6 a 



4. Find the volume of x 2 — y 2 = x 2 z 2 in the first octant, x varying 
from to h. Arts. \nh 2 

5. In Ex. 3 find the moment of inertia with reference to the 
z-axis. Arts. m\ln 2 

6. A rectangular parallelepiped which has its base in the xy- 
plane and its sides parallel to the other coordinate planes, 
intersects the hyperbolic paraboloid az = xy. Show that the 
enclosed volume = base X mean length of the vertical edges. 

7. Find £1 xy dx dy dz and £1 xyz dx dy dz for the octant of 



an ellipsoid. 



Arts. (1) 



a 2 b 2 c 



(2) 



a 2 b 2 c' 



1.3.5' v_/ 2.4.6 
8. Find the volume enclosed in the first octant by the ccor- 



(x \ i / v \ i / z \ % 
— ) + I ,) ■+ ( — ) =1. 



Ans. 



abc 







196 



INFINITESIMAL CALCULUS. 



[Ch. XXXV. 



180. Volumes. Polar coordinates. Let be the origin, 
OA the initial line, BAOC the initial plane. A plane re- 
volving through the angle <j> 
about OA from the initial plane 
contains a point P, OP = r mak- 
ing an angle 6 with OA. The 
polar coordinates of P are <£, 0, 
r; r and 6 are the ordinary 
polar coordinates of plane 
geometry, and (§ 178) PM = 
r dd dr + Ii, where l\ is a higher 
infinitesimal. The increment 
dcj) brings P to T and PM 
to TQ, PT = PAd<f> = r am d<f>. 
Hence the element of volume PQ 




Fig. 111. 



= (r dd dr + /i)(r sin d<f> + I 2 ) = r 2 sin d0 dfl dr + / 3 . 



/. 7 



r 2 sin d<£ d# dr. 



(The element is, as in Fig. 110, PR. PS . P7 7 .*) 

Ex. 1. Volume of a sphere. (1) The origin being the centre, 

'2tt 



7 = 



o J 

•2k 











r 2 sin <i<£ <i# dr 



o 



o 



•Ja 3 sin 6 d<f> dd = 



2n 







fa 3 d<j> = %7za 3 . 



The first integral ^a 3 sin 6 d<j> dd = a, pyramid with vertex at the 
centre and base on the surface of the sphere, the second fa 3 d<£ = a 
wedge of which the angle is d$ and the edge a diameter of the 
sphere, the third f-;ra 3 =the sum of the wedges. 

(2) If the origin is on the surface, and the initial line a diameter, 



7 



2tt 
, 



2a cos0 



r 2 sin 6 d$ dd dr. 



ISO.] 



SUCCESSIVE INTEGRATION. 



197 



(3) If the origin is a point on the surface, the initial line a tan- 
gent, and the initial plane passes through the centre, 



7 = 4 



7C 7Z 

'2 "2a sin 6 cos d 



0. 



r 2 sin 6 d<f> dO dr. 



o 



2. The vertex of a cone of vertical angle 2 a is on the surface of 
a sphere of radius a, and its axis passes through the centre of 
the sphere. Find the common volume. Arts. |-7ra 3 (l — cos 4 a). 

3. The moment of inertia of a sphere about a diameter 



C2i 



J 



„ 



ixr 4 sin 3 # d<j> dd dr=m fa 2 , 



and about a tangent line 



= 4 


°~2 


7C 

"2 


• 


. 


J 



2a sin cos <f> 



jjtr 4 sin 3 # d<£ dd dr = m ia 2 . 



o 



4. The potential of a solid sphere (density p) at a point on the 
surface = 



f2w 





J v J 



2a cos 772- 

jj.r sin d<£ d# dr = f ^7ra 2 =~. 
o 



a 



5. To find the potential 7 of a spherical shell of infinitesimal 
thickness (radius r, thickness t, density //) at any point A. 

Take 0, the centre of the sphere, as origin, OA as initial line, 
and let 0A = c. Then 



7 = 
7= 



r27r 



> jj.r 2 sin d d<f>dd .t 



o 
2nfj.rt. 



and 



2nprt 



oV / c 2 + r 2 — 2cr cos 
[(c + r)-(c-r)], c>r, 



, where r is constant. 



and 



[(r + c) — (r— c)], c<r. 



V = = — for A outside the sphere, 

c c 

171 

= ^7rfJtrt = — for A inside the sphere, 
r 









^M 



198 



INFINITESIMAL CALCULUS 



[Ch. XXXV. 



In the former case the potential is the same as if the whole 
mass were at the centre of the sphere, in the latter it is inde- 
pendent of c, and therefore the same at all points inside the sphere 

6. To find the potential of a homogeneous solid sphere (radius a) 
at any point A. 

Consider the sphere as made up of concentric shells of infini- 
tesimal thickness t = dr. Then from Ex. 5, 



C 



« 4 TTfia 3 m . 

r 2 dr = ^- = — , A outside, 

r\ O C C 



and 



Aufi 



r 2 dr + 4:7zfi 



a 

r dr = 2nii{a 2 -lc 2 ) J A inside, 



181. Volume. Mixed coordinates. A volume = 



zdxdy 



if the base is in the ^-plane. Instead of dx dy for the tase 

of the column of height z we 
may take the polar element of 
area r dcf> dr. Then 




V= 



zr d<fi dr. 



Ex 1. For the volume of a 
sphere by this method, 



XV — 



V = 



■2* 



Ca 



Fig. 112. 



o 





Va 2 —r 2 r d<f> dr 



o 



ia 3 d<f> = %na z . 



The first inetgral \a z d<f> is a wedge whose edge is OZ and angle 
d<f>, the second is the sum of such wedges for a hemisphere. 

2. The axis of a right circular cylinder of radius b passes through 
the centre of a sphere of radius a(> b). Find the common volume. 



•2n 







Cb 



Va 2 -r 2 rd<}>dr = %n[a 3 - (a 2 - b 2 )*\ 



181.] 



SUCCESSIVE INTEGRATION. 



199 



3. A right circular cylinder of diameter a penetrates a sphere 
of radius a, the centre of the sphere being on the surface of the 
cylinder. Find the common volume. 



TZ 

f] 

Jo J 



'a cos <f> 



Va 2 — r 2 rd<j>dr = l (37r — 4)a 3 . 



4. The axes of two equal right circular cylinders of radius a 
intersect at right angles. Find the common volume. 



8 



71 

J 



"2 /l — COS 3 ^ 



Va 2 — r 2 sin 2 0r d<b dr = %a 3 ( — r—r-, — ) d<b 
o Jo ^ sm '* / 

it 

= fa 3 tan — + sin <f> =-V 6 a 3 . 



5. The volume between the surface z = e~( x2 + ^ 2 ) and its asymp- 
totic plane 2 = 0. 

This is a surface of revolution about the 2-axis 



zrd<f>dr, and x 2 + y 2 = r 2 . 



(1) F= 

• 


•2tt 
. 


roo 

zr d 


^ 


.-. V = 2tt 


i 


(2) Also 


V 


= 4 

• 


00 





n 00 

L J 



|« 00 |«00 

J V 



oo 



2 dx dy 



e~y 2 dx dy 



= 4 



'00 



roo 



6-2/ 2 C?2/ • 



/•OO 



er x2 dx* = 4: 



o 



6~ x2 C?X ) . 



JO 



.\ From (1) and (2), 



'00 



e~ x2 dx 



1Z 



(Cf. § 124.) 



o 



* If the integration limits of each variable are independent of the 
other we evidently have 

J Jf(x)F{y)dxdy = Jf(x)dx JF{y)dy. 



200 INFINITESIMAL CALCULUS. [Ch. XXXV. 

182. Area of any surface. Let the parallelepiped. Fig. 42, 
which has the base dx dy in the #?/-plane and its sides parallel 
to the 2-axis, intersect the tangent plane at P{x, y, z) and 
the surface in sections of area A t and A s , respectively, which 
are assumed to be equivalent infinitesimals. Let a, /?, y 
be the direction angles of the normal at P. Then 

dxdy = A t cos y, .". A s = sec y dx dy + 1, 

where / is a higher infinitesimal. Hence the surface S 

= sec y dx dy= sec a dy dz = sec /? dz dx. 

uu uvl 3i^ 
cos a y cos B, cos y are proportional (§ 63) to ^— , ^-, ^— if 

r ' 9z ay 3-2 

3z 
m=c is the equation of the surface, and hence to 1. — ^— , 

do; 

3z 

if the equation is in the form z = f(x, y). Hence 



dy 



sec y 



or 



sKlt 


/3u\ 2 


/9w\ 2 
A9J/ 


3m 
dz 


.U/9 


Z\ 2 /32' 


\ 2 



% 



Ex. 1 A right circular cylinder of diameter a penetrates a 
sphere of radius a, the centre of the sphere being on the surface 
of the cylinder. Find the surface of the sphere which is inside 
the cylinder. 

Let the equations be x 2 +y 2 +z 2 = a 2 (1), x 2 +y 2 = ax (2). 

For (1) sec r = ~ = / =:- 

z Va 2 -x 2 -y 2 



182.] 



SUCCESSIVE INTEGRATION. 



201 



.\ S = 4 PT 
JoJo 



a Cv'ax-x* a dx dy 



Va 2 



= 4a (x + a) sin -1 , 



x'-y 

'a+x 



- = 4a sin -1 * I — ; — dx 
< 2 Jo \a+x 



* 



as a = 2(^-2)a 2 . 
-»o 



2. In Ex. 1 find the volume of the cylinder which is inside the 
sphere. 



^a 



a 



For (2), sec/? = — = ■ — . — , and (1) and (2) intersect in 

V 2Vax—x 2 



z 2 = a 2 — ax. 



.\ £ = 4 



Wa 2 -ax a dx dz 



J 







2\/ax — x 2 



= 2a 



a \2l 
«\x 



dx = 4a 2 . 



3. The axes of two equal right circular cylinders of radius a 
intersect at right angles. Find the cylindrical surface enclosed. 
Let the equations be x 2 + z 2 = a 2 } y 2 + z 2 = a 2 . Arts. 16a 2 . 



*Lets/(a+aO=sin 2 0. 



™^ 




CHAPTER XXXVI. 



MEAN VALUES. 



183. Let the base b— a of the curve, Fig. 113, be divided 
into n equal parts, at the extremities of which ordinates 

2/i? V2, • • « are drawn. The 
limit of 

3/1+2/2 + - - , +Vn 

n 

for n infinite is called the 
mean value of y for the in- 
terval a to b of x. If dx = the 
length of the equal segments 
of b— a, n=(b—a)/dx. Hence 




Fig. 113." 
the mean value of y is 



rb 



£Zy 



dx 



y dx 



a 



a 



a) 



Since the numerator = the area A BCD, the mean ordinate 
is equal to the height of a rectangle which has the same 
base and area as the given figure. 

The result (1) may be regarded as the mean value, for 
the interval a to b of the variable, of any function which is 
single- valued and continuous for that interval. 

Ex. 1 The mean ordinate of a semicircle of radius a = ^-— =- 

2a 4 

= • 7854a. 

202 



183, 184.] MEAN VALUES. 203 

2. The mean square of the ordinate of a semicircle 



a 



y 2 dx 



a 



(a 2 — x 2 )dx 



— a _J —a _ 2 n 2 



2a 2a 

3. Find the mean ordinate and the mean square of the ordinate 
of the curve y = a sin nx from x = to x = n. 

Arts. (1) 2a/ri7: } (2) a 2 /2n. 

4. The arc of a semicircle is divided into equal parts, from 
the extremities of which perpendiculars are drawn to the diameter. 
What is the mean value of their length? 

Radii through the points of section divide the angle at the 
centre into equal parts. Hence 



. n re a 
£ la sin 6-7- — = — 

dd 7C 



sin dd = — . 

?! 



5. In Ex. 4 find the mean distance of the points in the cir- 
cumference from one end of the diameter. Arts. ^a/n. 

6. A straight line A B of length a is divided into equal parts, 
P being a point of section. Find the mean value of AP . PB for 
all positions of P. Arts. \a 2 . 

7. The northern hemisphere is divided into zones of equal 
area. Find their mean distance from the pole. 

Ans. Arc = radius. 

8. A rectangle is divided into rectangles by lines which divide 
the sides equally. Find the mean square of the distance of the 
rectangles from one corner of the given rectangle. 

The sides at that corner being axes, 

ab 1 C a t h 
£ I ^ 2 + y 2 )^-^^ y = - b I (x 2 + y 2 )dxdy = Ua 2 + b 2 ). 

184. Let dX be an element of any quantity (length, area, 
volume, mass, time, etc.), and u a variable which is taken 
any number m times per unit of X. Then 

£1 um dX £1 u dX 
£1 m dX = £1 dX 



204 INFINITESIMAL CALCULUS. [Ch. XXXVI. 

expresses the limit of the sum of the u's divided by their 
number, and is thus the mean value of u for the range in- 
volved in the summation. If the elements are unequal 
the result is still the same as the mean value of u taken 
once for each of the elements if they were equal, since 1 for 
each one-mth part of the unit is equivalent to m per unit. 

Ex 1. To find the mean distance of points within a circle of 
radius a from a given point on the circumference. 

In this case eU is an element of area, say r dd dr (§178), u 
is r, hence the mean value 



2" f 2 

Jo 



2a cos 

r 2 d6 dr 

32 
- =— a. 



■KO? 971 

2. The plane base of a hemisphere of radius a is horizontal. 
Find (1) the mean height of points within the hemisphere (the 
element being one of volume), (2) the mean height of points on 
the curved surface, (3) the mean depth of points in the base 
below the curved surface. Ans. (1) fa, (2) \a, (3) fa. 

In (1) and (2) the mean height is the height of the centre of 
gravity,* in (3) it is volume/base. 

3. Find the mean square of the distance of points within a 
sphere of radius a from (1) the centre, (2) a point on the surface, 
(3) a diameter. Ans. (1) fa 2 , (2) fa 2 , (3) fa 2 . 

* The point whose coordinates are the mean values of the rec- 
tangular coordinates of points in a body for equal elements of mass 
is easily seen to be the centre of mass (or of gravity) and therefore 
the centroid if the body is homogeneous. The centroid is sometimes 
called the centre of mean position. 



CHAPTER XXXVII. 



INTRINSIC EQUATION OF A CURVE. THE TRACTRIX 

THE CATENARY. 

Intrinsic Equation of a Curve. 

185. Let the tangent or normal of a curve turn through 
an angle X while the point of contact moves a distance s 
along the curve. The equation con- 
necting s and X is called the intrinsic 
equation of the curve. 




Ex. 1. The intrinsic equation of a circle of 
radius a is obviously s = aX. 

2. To find the intrinsic equation of the 
semi-cubical parabola ay 2 = x 3 (Fig. 29), the Fig. 114. 
intrinsic origin being the cusp. 

The tangent at the origin is the z-axis. Hence 

, d y 3 / x \ i 
ax 2 \a/ 

.*. x=$ata,n 2 X, whence y = ?\a tan 3 X. 

Also ds = Vdx 2 + dy 2 = fa sec 3 A tan X dX. 

/. s = |a sec 3 A tan X dX, or s^-fca (sec 3 X — l), 
Jo 

the equation required. 

3. Find the intrinsic equation of the common parabola y 2 = 4ax, 
the origin being the vertex 

If the tangent make an angle X with the positive direction of the 
?/-axis, tan X = dx/dy. 

Ans. s = a sec X tan X + a\og (sec A + tan X). 

205 



206 INFINITESIMAL CALCULUS. [Ch. XXXVII. 

4. Find the intrinsic equation of the cycloid x = a(0 — sin 6), 
y = a(l — cos 6) (Fig. 19), the origin being at a cusp (say, at x = 0, 
2/ = 0). 

[Show that X( = YSP) = id.] Arts, s = 4a(l - cos X.) 

5. Find the intrinsic equation of the four-eusped hypocycloid 
x = asin 3 d, y = a cos 3 # (Fig. 18), the origin being at a cusp (say at 
x = 0, y = a). 

[Show that X( = OST) = 6.] Arts. s = fa(l -cos 2X). 

6. Show that an equation may be transformed to a new origin 
at (s f , X') by substituting s + s' for s and X + X' for X m the given 
equation. 

7. Find the equations of the curves of Exs. 4 and 5 when the 
origin is a vertex (the middle point of the arc between successive 
3usps). Ans. (l)s = 4asin^, (2) s = %a sin 2X. 

186. Instead of expressing x and y in terms of X, we 
may be able to express s and X in terms of x or some other 
variable, and eliminate that variable. 

Ex.1. The catenary y = a cosh (x/a) (Fig. 117), the origin 
being the lowest point, and hence the initial tangent being parallel 
to the a:-axis. 

tan X = dy/dx = sinh. (x/a), and c?s = cosh (x/a) dx, whence 
s = a sinh (x/a). / . s = atan^. 

2. The cardioid r = a(l— cos 6), the cusp being the polar and 
the intrinsic origin. 

ds = Vr 2 d6 2 + dr 2 = 2a sin £0 dd, .*. s^=4a(l-cos $0). 

Also X == 6 + </>, where ( § 136) tan cp = r dd/dr = tan \d. 
:. A=f0. Hence s = 4a(l — cos \X). 

Show that the equation is s = 4a sin \X if the origin is the point 
most remote from the cusp. 

3. Show that the intrinsic equation of an epicycloid is 

4b(a + b) / aX 



a 
when the origin is a cusp, and 



/ a/ \ 

1 -cos — — ) , 
V a + 26/' 



4b(a + b) . aX 
s = — — - sin 



a a + 2b' 

when the origin is a vertex. 




186-189.] THE TRACTRIX. 207 

187. The radius of curvature. The curvature is the 

s-rate of a (§ 86) and hence R = ds/dX = f(X) if s = f(X) is 
the equation of the curve. 

188. The evolute. Let be the origin of the given curve 
s = f(X), P any other point on the curve, C, Q the centres of 
curvature of and P, and let C be taken 
as the origin of the evolute. Then PQ 
= /'(A) and OC = /'(0), also CQ = PQ- OC 
(§ 90 (2)). Hence the equation of the 
evolute is 

s=/'W_./'-(0). 

Ex. 1. Show that the evolute (1) of a parabola is a semi- 
cubical parabola, (2) of a cycloid is an equal cycloid, (3) of a 
four-cusped hypocycloid is a similar curve of twice the size of 
the given curve, (4) of a cardioid is a cardioid of one third the 
size of the given one. 

2. What is the intrinsic equation of the involute of (1) a circle, 
(2) a catenary, the involute beginning at a point on the given 
curve? Ans. (1) s = iaP, (2) s = a log sec L 



The Tractrix. 

189. This is the curve in which the tangent is of con- 
stant length. 

Let {x, y) be the coordinates of a point P on the curve 
(Fig. 116), and let the tangent PT = a* 



From the figure dy/dx= — y/Va 2 —y 2 , from which the 
equation may be found by integrating, the result being 

x = a sech -1 (y/a) — V a 2 — y 2 . 



* The curve is the path of a body which is drawn along on a rough 
horizontal plane by a string of length a, the other end of which is 
moved along a straight line OX; whence the name of the curve. 



208 



INFINITESIMAL CALCULUS. [Ch. XXXVII 



Since ydx= — dyV / a 2 —y 2 , the element of the area of the 
curve = the element of the area of the circle of radius a, .'. 
the whole area between the curve and its asymptote (the 
#-axis) is the same as that of the circle, viz./7ra 2 . 




Fig. 116 



The length of the curve from Y to any point whose ordinate 
is b may be found as follows: 



ds 
dy 



— ~~~" < • • & — ti> 



a y \o/ 



Let the tangent make an angle X with YO. Thens = 
a log sec A, which is the intrinsic equation (origin Y) of the 
curve. Hence the radius of curvature = ds/dX = a tan X, 
and the evolute is (§ 188) s = a tan J, a catenary (§ 192). 

The area of the surface of revolution * of the curve about 
the a>axis = 47ra 2 , and the volume = §7ra 3 . 



* This surface is known as the pseudo-sphere. 



190-192.] 



THE CATENARY. 



209 



The Catenary. 

190. This is the curve formed by a uniform chain hanging 
vertically. 

Let A be the lowest point, P any other point. From P 
draw PB vertically and equal to the length AP or s of the 
chain, and from B draw a horizontal line to meet the tan- 
ent at P in (7, and let BC = a. 







Y 












?x 






^^^^ 


' 


<K\ 






\c/ v 




( 


c,/ 


> \ 




B ^v 


A 




\K 





M 



Fig 117. 



N X 



191. Mechanics of the figure. The portion AP of the chain 
is in equilibrium under the action of three forces, viz., the 
horizontal tension at A, the tension at P in the direction of 
the tangent, and the weight, which is vertical. Hence PBC 
is a triangle of forces for AP, and since the vertical force on 
AP is the weight of a length PB of the chain, it follows that 
the tension at P is equal to the weight of a length CP of the 
chain, and the tension at A to the weight of a length a of a 
chain, and therefore a is constant. 

192. Geometry of the figure. Draw from A a vertical line 
and take QA=- a; take as the origin, OA as j/-axis and a 



210 INFINITESIMAL CALCULUS. [Ch. XXXVII. 

horizontal line OX as z-axis. Then OM = x,MP = y. (By 
this choice of axes the constants of integration will = 0.) 

Since CB, BP, and CP are a, s, and Va 2 + s 2 , respectively, 
we have 

dx a ds x/ a 2 + s 2 K " ds Va 2 + s 2 

From (3) dy=s ds/V a 2 + s 2 , .'. y=\ / a 2 + s 2 = CP, (4) 

.". the tension at any point P is equal to the weight of a 
length y of the chain. 



From (2), dx= ads/Va 2 + s 2 , .'. x=a sinb _1 (s/a), 

• /y* fi X X 

.\ s = asinh— . or — — (ea — e~~a), (5) 

a 1 

which gives the length from the lowest point to the point 
whose abscissa is x. 

From (1) and (5), dy = sinh — dx. 

(j/ 

x a i — —\ 

.'. y = a cosh — , or =-feo+e a), (6) 

the equation of the curve. 

From (1), s = a tan <f>, the intrinsic equation (origin A) 
of the curve. 

The normal. NP:MP: :CP:CB, .-. NP/y=y/a, or NP = 
y 2 /a. 

The radius of Curvature. R=ds/d<f> = a sec 2 <$> = ay 2 / a 2 = 
y 2 /a, or the radius of curvature is equal to the normal. 

Let D be the foot of the perpendicular from M on PT '. 
Since MP = CP, .'. MD = CB = a and DP = BP = s. 

Hence the locus of D is the involute of the catenary. 

Also MD, the tangent at D to the involute, is of constant 
length, .*. the involute Is a tractrix. 

The intrinsic equation of the e volute is s = a(sec 2 <fi— 1). 



CHAPTER XXXVIII. 
INFINITE SERIES * 

193. A series is a succession of terms which follow one 
another according to some law. The series is said to be 
infinite when it does not terminate. If we add the terms 
of the infinite series 1 + \ + \ + \ + . . . it is seen that the 
sum approaches a limit, viz., 2; for this reason the series 
is said to be convergent. That the limit is 2 is also seen 

by taking the sum of the first n terms, which is 2—^——^, 

and finding the limit of this when n= 00 . 

This series is a particular case of the infinite geometrical 
progression 

l+x + x 2 + x s + ... (1) 

From elementary algebra the limit of the sum is 1/(1 — a;), 
if |rc|<l. If \x\ > 1 the series has no limit or is non-con- 
vergent. 

194. Let ^0+^1+^2+^3+ . . . +u n + u n +i+ ... (2) 
be an infinite series , and let s n be the sum of the first n terms. 
The series is then convergent if s n has a limit when n = 00 ; 
let the limit, if it exists, be s. The limit must as in all other 
cases be a definite finite quantity (§ 2) such that s—s n has 
the limit 0. Hence a series cannot be convergent unless 
u n = when n=oo, i.e. unless the terms tend to a limit 0. 
This is a necessary but not a sufficient condition for con- 
vergence. 

*O y i the subject of Infinite Series the student may consult Osgood's 
Introduction to In-finite Series (Harvard University), also Gibson's 
Calculus (MacmillanV 

211 



212 INFINITESIMAL CALCULUS. [Ch. XXXVIII. 

If a series is non-convergent it is either divergent or oscilla- 
tory, divergent if s n becomes infinite with n, oscillatory if s n 
remains finite but does not approach a limit. Series (1) is 
divergent if |#|>1 or z—1, and oscillatory if x= — 1, s n in 
the latter case being 1, 0, 1, 0, etc., as n increases, but never 
approaching a limit. 

In general, a series is of no practical value unless it is 
convergent. 

195. Without expressing s n in terms of n it may be pos- 
sible to test a given series for convergence. For this pur- 
pose various methods are given in works on algebra; we 
recall a few results which are of importance in our work. 

(1) A series is convergent if (a) the terms are alternately 
+ and — , (6) the absolute value of the terms constantly 
diminishes, and (c) the limit of that value is when n=oo. 

Ex. l-i+J-i + ... is convergent (see §2, Ex.3). The. 
limit of the sum lies between '69314 and '69315. It = log e 2 (§ 197 
Ex. 1). The series 2 — f + £ — . . . satisfies conditions (a) and (6) 
but not (c). It is oscillatory (see § 2, Ex. 3). 

(2) The series 1+— +-—+... is convergent when c>l, 

Z c 6 C 

divergent when c<l. Thus the " harmonic series " 

1+1 

2 2 3 2 



l + i + i+ ... is divergent, but 1+^ + ^+ ... is conver- 



gent. 

11. 
Ex. 1. — — — + — — ~ + . . . is convergent, since each term is 

a 2 + l 2 a 2 + 2 & ' 

less than the corresponding term of 1 + — + . . . 

2. — -+ — - + ..., a> 0, is divergent. For it > 



a+l a+2 * ' ° 6+1 6+2 

+ . . . , where 6 is an integer larger than a, and this is a part of 
the divergent series 1 + \ + \ + . . . 

(3) If a series is convergent when all the terms are posi- 
tive, it will be convergent if any of the'terms are negative. 



195, 196.] INFINITE SERIES. 213 



cos x cos 2x 
p t 2 : 

if all the terms were positive they would not be greater than 

1 1 
1 



ill ^ / I i i ^ j^j i j 

Ex. -77- H — ^ — h... is convergent for all values of x } for 



the corresponding terms of — + Q ^ + . . . 



The contrary is not necessarily true, that is, a series may 
be convergent when some of the terms are negative, but not 
when those terms are made positive. Thus 1— i + J— ... is 
convergent, but 1 + J + J + . . . is divergent. 

A series is said to be absolutely convergent when the abso- 
lute values of the terms form a convergent series, i.e., when 
the series remains convergent if all the terms are made posi- 
tive. If not absolutely convergent it is said to be con- 
ditionally convergent. Thus 1 — i + i — 4"+ • • ■ -is absolutely 
convergent, 1— i + J— \+ ... is conditionally convergent. 
It is known that the convergence and limit of an absolutely 
convergent series are independent of the order in which 
the terms are taken, whereas the terms of a conditionally 
convergent series may be grouped so as to converge to a 
different limit (in fact to an assigned limit) or to diverge. 
For example, arrange the series 1— i+J— i+ ... as follows: 

(l~i) — i + ("5" — 6") — B" + (i — tV)"~ ••• 

This = i-i+£-i+ .•• =i(l-i+i-i+ ...), one-half 
of the original series. 

(4) A series uo + Ui + . . .+u n + u n +i +. . . is absolutely con- 
vergent if R, the absolute value of the limit of u n +i/u n when 
n= oo , is <1, divergent if R>1. It may or may not be 
convergent if R = 1 . 

196. Power series. The most important infinite series are 
those of the form 

ao+aiX-\-a 2 x 2 + a s x 3 + . . . +a n £ n + a n+]L £ n+1 + . . . , (1) 

which is called a power series in x. The indices are posi- 
tive ascending integers, and ao, a\ } . . . are independent of x. 



214 INFINITESIMAL CALCULUS. [Ch. XXXVIII. 

By assigning values to x any number of series may be formed 
from a given power series. 

Let the absolute value of the limit of a n /a n +i when n=oo 
be r. It follows from § 195 (4) that the power series is 
absolutely convergent if |#|<r- (i.e. if x > — r and <r) 7 
divergent if |x|>r. If x = r the series may be convergent 
or non-convergent. If a n /a n +i = co when n=oo,the series 
is convergent for all values of x. 

Since the limit for n + 1 = oo is the same as for n = oo , 
the value of r may be found equally well from a n+ i/a n+2 
or from a n _i/a n , or any two successive coefficients. 

Ex.1. 1+X+—+- + . ..+—+.. . 
2 3 n 

1 o»n + l'l, 

dn = —, .*. — -= =1 + — =1 when n = oo, .\r = l. 

Hence the series is absolutely convergent if |x|<l. It is con- 
ditionally convergent if x— — 1, divergent for all other values of x. 

/y» 2 /v»3 1*71 

2.- l+z+— +— +. . .+ , + . . . 
2 ! 3 ! n ! 

dn/dn+i = (n + 1) ! /n ! = n + 1 = oo when n = oo . 

Hence the series is absolutely convergent for all values of #. 

3. £ +— ■}-■— + . . . and 1 + + — + . . . are parts of the series of 

Ex. 2, and are therefore absolutely convergent for all values of x. 

a mi t^- -to.- ■* ra(m — 1) , 

4. Ine Binomial Series l+mx + — ar + . . . 

— i 

an/an+i==(n + l)/(m — n) = — 1 when n = oc y .*. r = l. 

Hence the series is absolutely convergent if \x\ < 1 whatever the value 
of m. It may be proved to be absolutely convergent if \x\ = 1 and 
m is positive, and conditionally convergent if x== 1 and — 1 <m <0. 

5. If the power series a -{- a x x + a 2 x 2 -\- . . . is convergent when 
|x|<r, the series a 1 +2a 2 x+ . . . formed from the derivatives of 
the terms is also convergent when \x\ <r. 

For / -^— — = r ^-\ = \r 

^- fl0 n + l'on+ 1 ^"-"an+S ' 



197.] INFINITE SERIES. 215 

6. Show also that the series c + a x + %a 1 x 2 + ^a 2 x 3 + . . . formed 
by integrating (c being a constant) is convergent if \x\ <r. 

197. A power series a +aix + a 2 x 2 + . . .may be such 
that the limit of the sum for every value of x within the 
interval of convergence (—r<x<r) is equal to the value 
of some function }(x) for that value of x. On this under- 
standing we may write 

f(x) =do +CLiX + 2 2 X 2 + . . . (1) 

Thus if |z|<l, (l+x) m = l+mx+-^-— — -x 2 + . . . , and 

as a particular case 1/(1 + x) =1 — x-\-x 2 — .... If |#|>1 
there is no connection between the value of the function 
and the sum of the terms of the series. 

If we differentiate both sides of (1) as if the series were 
finite we have 

fix) =ai+2a 2 x + 3asx 2 + ... (2) 

Similarly, multiplying (1) by dx and integrating, 

F(x) =c + aox + \a\x 2 + \a 2 x z + . . . , (3) 

c being the integration constant, viz., the value of F(x) 
when x=0. It may be proved that these results hold true 
(i.e., the new functions are equal to the limits of the sum 
of the terms of the new series) for all values of x for which 
the new series are convergent. These values are — r<x<r 
(§ 196, Exs. 5, 6), the same as those of the original series. 
If, however, any one of the series is also convergent when 
x=r or — r, it must not be assumed that the others are also 
convergent for those values. Integration in general increases 
the rapidity of the convergence of a series, and it may change 
a series which is divergent when x = r into one which is con- 
vergent. 



216 INFINITESIMAL CALCULUS. [Ch. XXXVIII. 

Thus l + %x + ^x 2 + ... is divergent when x = l 9 but 
x+7^z 2 + 7^x s + ... is convergent when x = l. 

Ex. 1. Logarithmic series. If \x\ <1, 

: = 1— x + x 2 — x* + . . . 

1+x 

Multiply by dx and integrate. Then if |rc| < 1, 

log (l+x) = x-^x 2 + ^x 3 -ix* + . . . (1) 

(No constant of integration, since both sides vanish with x.) 
(1) is also convergent if # = 1, § 195 (1), .'. log 2 = 1— $"'+i~~ . . . 
(1) may be used for the calculation of the Napierian logarithm 
of any number > — 1 and < 1, but it converges too slowly to 
be of much value for such calculation, unless \x\ is very small. 
Change x in (1) into —x and subtract from (1). Then 

lo S (fz|) =2(x + ix 5 +ix' + . . . ). (2) 

Let y = (l+x)/(l— x), then x= (y — l)/(y + l) Substituting 
in (2), 

This series may be used for the calculation of any Napierian 
logarithm, since (y — l)/(y + l) is necessarily a proper fraction 
when y is any positive quantity. 

Thus if y = 2, (y-l)/(y + l) = h 

.-. log e 2 = 2[Kia) 3 +i(i) 5 + ...]='693147. 

Similarly log e 3 = 2 [i + i(i) 3 + s(i) 5 + . . . ] = 1*098612. 

Also, loge 4 = 2 log e 2 = T 386294. 

Another series may be derived from (2) thus: put (l-\-x)/(l—x) 
= (l+y)/y, then x = l/(l+2y); hence, remembering that 
log [(1 +y)/y] = log (1 +y) -log y, 

log(l +y ) = log y + 2[ 1 -^ + j( 1 -^) 3 + |( f ^) 5 + ...]. 



179.] INFINITE SERIES. 2l7 

Thusif ?/ = 4, loge5 = loge4 + 2[i4-KI) 3 +*(i) 5 "f ..] = r609438. 
Hence loge 10 = log e 5 4- loge 2 = 2*302585, and hence the modulus* 
of the common logarithms (which = l/log e 10) is '4342945. The 
common logarithms may therefore be found from the Napierian 
logarithms by multiplying by '4342945, and, conversely, the 
Napierian from the common logarithms by multiplying by 2*302585. 

2. Gregory } s series. Prove that for — 1<#<1, 

tan -1 x = x — Ja* 3 + Ja* 5 — . . . (1) 

This series gives the radian measure of an angle in terms of 
its tangent. Show that 

! = i_ KW+ ...= 2 ( I L + _L + ...). 

This converges slowly, but by applying (1) to the relation 

Tt 1 

— = 4 tan -1 \ — tan -1 — - a rapidly converging series for the calcu- 
lation of n is obtained. 

TT 1 . , 

Since tan~ 1 o* = — — tan -1 — , if b > 1 we have 
2 x 

, TT 1 1 1 

tan—a* =— ^7r^~^-E + ' • • 

2 x 3a* 3 5x 5 

3. Prove that 

1 x 3 1 .3z 5 1 .3. bx 1 . . " 



and hence, making x = -|, that 

1 3 

24 + 640 H ' 



^K 1+ 4 + ^ + 7i8 + --) =3 ' 14159 --' 



* If x = \og a y, then by definition of a logarithm, a x = y. Taking 
logarithms of this, the base being supposed e, we have x \og e a = \og e y, 

,\ x or logay = 1 sc , or the logarithm of 2/ is changed from base e 

to base a by multiplying by = , which is called the modulus of 

the system of base a. 

If y = e. logo e =? , hence the modulus of the common system 

* ' & log e a J J 

is also equal to log l0 e. 



218 INFINITESIMAL CALCULUS. [Ch XXXVIIL 

Show also that sec _1 a; = — — — r— - — * ' , — . . . , b]> 1. 

2 a; 6x 3 2.4.5r ' v ' 

198. Maclaurin's Series. If there is a power series * 
which =/(s), that series is /(0) + f\0)x + J -^x 2 + . . . 
For, suppose it to be ao + aiX + a 2 x 2 + . . . Then 

}(x) =do +CLiX-\-a 2 X 2 + CL3X 3 + CL4 : X 4: + ... (1) 

Differentiating successively, 

/ / (x)=a 1 -! 2a 2 x+3asx 2 + 4:a4 : x 3 + ... (2) 

/" (x) =2a 2 + 2. 3a s x + 3 . 4a 4 x 2 + . . . (3) 

}'"(x) =2 . 3a 3 +2 . 3 . 4a 4 z + ... (4) 

etc. If (1) is convergent for |x|<r, (2), (3), . . . are con- 
vergent for the same values of x, and this interval includes 
x = 0. Making x = in (1), (2), (3), . . . we have 

/(0) = oo, f(0) = a 1 , /"(0) = 2a 2 , /'"(0) = 2 . 3a 3 , . . . 

Substituting in (1), 

f"(Q) f'"(0) 
f {x) = m+r{0)x + ^ x 2 + l^ x s + . . . (5) 

This is Maclaurin's Series. In substituting in (5) we are 
said to expand or develop }(x) into a power series. 

Ex. 1. To expand sin x. 

f(x) = sinx, .'. /(0) = sinO = 0; 

f'(x) = cosx, .'. /'(0) = cosO = l; 

f"(x)==-smx, /. /''(OH -sin = 0; 

/'"(a) = -cos a, .*. /'"(0) = -cos 0= -1, etc. 



Substituting in (5) we have 



x 3 x* 



smx = x-— +Fj — • • • , (6) 

which gives the sine of an angle in terms of its measure in radians 

* There is such a series in most cases if f(x) and all its derivatives 
are real and finite when x = 0. See § 210. 



198, 199.] INFINITE SERIES. 219 

The expansion of cos x may be found in the same manner, or 
by differentiating (6): hence 

coss = l- — + —-... (7) 

(6) and (7) are convergent for all values of x (§ 196, Ex. 3). 

2. The expansions of e x and a x are particular cases of Maclaurin's 
Series. For 

f(x) = e x , f'(x) = e x , f"(x) = e*, etc.; 

.-. /(0) = 6° = 1, /'(0) = 1, /"(0) = 1, etc. 

.-. e*=l+x + '- +- + ... (8) 

/I 2/v»2 /J 3/v» 3 

Similarly, a x = l+Ax + — r + -^—- + . . . ( A = log e a) . (9) 

These series are also convergent for all values of x (§ 196, Ex. 2). 

199. If we attempt to expand cot x by Maclaurin's Series 
we meet a difficulty at the outset, viz., cot = 00. This 
implies that there is no power series for cot x. 

Ex. To expand x cot x, 

2 x A 



/„ X" X* \ X L 

V-^JC--) 1-51+ 



cos a; 
x cot x = x— = 



sin x x z x 5 x 2 x* 

3! 5! 3! 5! 

Assume x cot # = 1 + a 2 x 2 4- a A x 4 + . . . (There cannot be any 
odd powers, since x cot x is an even function. See Ex. 12, p. 40.) 
Then 

1_ 2i + 4!~' " = \ 3! + 5l~* ' 7 ( l + W 2 + W* + . . . ). 
Multiplying * and equating coefficients, 



Hence 



x cot X = 1 - 


X 2 

~3~~ 


4 
~45~ 


1 


X 


X 3 


tut U/ 

X 


3 


45 



* The product of two convergent series is obtained by multiplying 
them term by termas if they were finite series, provided that one 
of them (at least) is absolutely convergent. 



220 INFINITESIMAL CALCULUS. [Ch. XXXVIII 

200. Formulae derived from the exponential series. In 

§ 198, (8), write xi for x, where i=v — 1 (observe that i 2 = — 1, 
i 3 = — V^T, i 4 =l, t 5 = ^ etc.). 

•* V 2! + 4! '• 7 + H 31 + 51 'V' 

or e 2 * = cos # -H sin z. (1) 

Writing — z for a:, 

e~ xi = cos #— i sin x. (2) 

Adding and subtracting, 

cosx = %(e? i + e-~ xi ), (3) 

sin #=— (e 2 *— e~**), (4) 



whence 



1 ( e xi ~e~ xi \ 1 / e 2xi - 1 \ 
tan ^-J^ + e-^/ ~7 \^+l) ' (5) 

These are Euler's Formulae. 
In (1) write nx for x. 

.*. cos nx+i sin nz = e n:ri = (e**) w = (cos x+i sin #) n t 

.' . (cos x + i sin #) n = cos nx+i sin nx. (6) 

This is Demoivre's Theorem. 

201. Taylor's Series. Suppose the function of x to be 
f(h + x). The first ^-derivative is 

f'{h+x)d(h+x)/dx=f(h+x). 

The second is f"(h + x), etc. The values of the function 
and its derivatives for x = are f(h), f{h), /"(/&)> . . . Hence 
Maclaurin's Series takes the form 

j(h + x) = f(h)+f\h)x + } ^X 2 + } ^^x3 + . . . 

This is Taylor's Series. The conditions under which 
the function is represented by the series will be considered 
in Ch. XXXIX. 



200, 201.] 



INFINITE SERIES. 



221 



jYlitYl — 1 ) 

Ex.1. (h + x) m = h m +mh m - 1 x + - — — h m ~ 2 x 2 + ... 



2! 



sin h . cos ft 



X' 



2. sin (ft + z) = sin ft + cos ft . £ 

Z I o I 

3. If /(#) = x 3 — 2x 2 — x + 3, writedown f(x + h). 



X i" • • • 



Examples. 

1. Expand (l+#) m , log (1+z), tan -1 x, by Maclaurin's series. 

a: 3 2x 5 llx 1 

2. tan*-* + — +— +^- 9 + ... 

, z 2 5x 4 61z 6 

3. sec x = l+- + — + — + ... 



a: 2 x 4 



or 17ar 



4. log sec x = ---\ 1 — . 

6 2 12 45 2520 



+ . .. 



5. cos 3 # = l 



3x 2 7x 4 



8 



„ H X 2 X* T 

6. e*™* = l+x+— —^-tz + -- 

2i o 15 

7. e x sec x = l+x-\-x 2 + %x 3 + . . . 

1 x 7x 3 

8. coseca;=-+— +t— — 

a: 3! 3.5! 

9. Show that 



+ 



x 2 x* 



X 3 X 5 



cosh x = l +—+— + . . . , sinh x = x+— +— + . . . , 
2! 4! o! 5! 



and hence that cosh x = cos ix, sinh #= — i sin tz, where i = v— T 

^ A .111 7Z 2 t 

10. Assuming — +—+— + . . . = — , show that 
V 2 2 3 2 6 



1 1 



"' 



p + 3i + 5- 2 + --=8' and 



1 _1_ 1 

p 2 2+ 3" 2 



12' 



Also that 



f 1 ! 



1 7T* 

J log (l+a:)dx = Y2. 



ri 



11. Show that 



cfc 



_ _1_ 1 L .; 



_ o^l+x 4 2 5 2.59 

12. When x = show from the series that 



(1) (sinaO/z=l, (2) (tanx)/s= 1, (3) (1-cos z)/:r 2 =ss J, 
(4) (tanx — sinx)/x 3 = J, (5) (e* — 1)A=1. 



222 INFINITESIMAL CALCULUS. [Cxi. XXXVJII. 

13. If a circular arc (radius a) subtends an angle 6 at the centre, 
show that when 8 is very small 

arc — chord = 4t a ^i nearly. 

14. If 6 is a small angle, show that 

sin d = dVcos~6, ] 

. , }■ nearly. 

tan 6 = 6^/cos~'0 J 

From these formulae are derived the rules given in Mathe- 
matical Tables for finding the sines and tangents of small angles. 

15. The chord of a circular arc is C, the chord of half the arc 
s c; show that the length of the arc is 

2c + J (2c — C), very nearly. 

This formula (Huyghens's) will give J of the circumference 

of a circle of 100 feet radius with an error of less than 1J inch; 

it gives i of the circumference of the same circle with an error 
of less than -fa of an inch. 



CHAPTER XXXIX. 



TAYLOR'S THEOREM. 



202. In Ch. XXXVIII the existence of a power series 
for j(x) or f(h + x) is assumed. We have now to consider 
under what circumstances this assumption may be justified. 

203. Theorem of Mean Value. Let f(x) be a single-valued 
function, and suppose f(x) and its first derivative /'(#) to 
be continuous from x = a to x = b. The Theorem of Mean 
Value asserts that 



b— a 



=/'(^i), 



a) 



where xi is some value of x 'between a and b. 




a c B 
Fig. 118. 




B X 



Let (Fig. 118) 04 = a,0B=b. If PRQ represents the 
graph of f(x) from x = a to x = b, AP = f(a), BQ = J(b). Hence 

'-±-4 — -^- is the slope of the straight line PQ. At some 
b—a 

point R between P and Q the tangent is parallel to PQ, and 

the slope of the tangent at R is /'(#i), where x\ = OC. 

Hence the equality stated in (1). It should be noticed 

(Fig. 119) that X\ may have more than one value. 

223 



224 



INFINITESIMAL CALCULUS. [Ch. XXXIX. 



The theorem may not be true if, between x = a and x = b, 
f(x) (Figs. 120, 121) or f(x) (Figs. 122, 123) has a finite 
or infinite discontinuity. 






Fig. 120. Fig. 121. Fig. 122. Fig. 123. 

204. If, in (1), /(&) = /(a), then / / (a;i) = 0; i.e., if f(x) and 
f'(x) are continuous from x = a to x = b, and if /(a) = /(&), 
then f'(x) = for at least one value between x=a and x = b. 
This is known as Rollers Theorem. 

205. In Fig. 118 let 4B = A and AC = 0h, O<0<1. Then 
(1) becomes 

f(a+h)-f(a) 



h 



= f(a + 0h), 



or 



f(a + h) = f(a)+hf'(a + Oh). 



This may be regarded as the beginning of an expansion 
of }(a + h) in powers of h. We have now to show that the 
expansion may under certain circumstances be continued 
to three or more terms. 

206. Taylor's Theorem. Let f(x) and its first n deriva- 
tives be continuous from x=a to x=a+h. Let P be a 
quantity which is such that 



/(a + 7i)-/(a)-/ / (a)^-Q^/i 2 ~ . . . 



2! " '" ,(n-l)! 
Consider also the following function of x: 

fix) 



f(n-l)( a ) 



f(a+h)-f(x)-f(x)(a+h-x) 



2! 



-(a-t-h— x) 2 — . . . 



(n- 1) 1 



{a+h-x) n - 1 -P(<i+h-x) n , 



(3) 



204-208.] TAYLOR'S THEOREM. 225 

and let this be called F{x). Differentiating, we obtain 

F\ x )=- j {n ^\ Aa+h-x)"-i+Pn(a+h-xy- 1 . (4) 
(n— 1)! 

Since f(x) . . . / (n) (x) are continuous from x =a to x =a-\-h, so 
also are F(x) and F'(x). Also F(a) =0 by (2), and F(a + h) =0 
by (3); hence (§ 204) F'(x) =0 for some value a + dh between 

a and a+h. Hence, from (4), P=— 



n\ 



Substituting in (2) and transposing, 



f"(d) fa-Vta) 
f(a+h)=f(a)+na)h+!-^hz + . . .+L W ftn -i 

! £ [71 — l) I 



f (n) ^ d %«. (5) 



n! 

Hence if f(x) and its first n derivatives are continuous 
from x=a to x=a+h, f(a+h) can be expanded into the 
finite series (5). This is Taylor's Theorem. It is really a 
generalization of the Theorem of Mean Value. It should be 
noted that h is not necessarily positive. The number 6 
>0 and <1, but its value generally depends upon a, h, 
and n, as well as the form of the function. 

Ex. log { a + h) = loga+~- 2 + --. ..+ n{a + eh)n - 

207. If x is written for a, (5) takes the form 

f(z + h)=f($+r(x)h+^h*+ . . . + / (n) ^+ gft ) ftn, 

208. The remainder. The last term of (5) is known as 
Lagrange's form of the remainder (R n ) after n terms. Another 
form of the remainder R n (Cauchy's), viz., 

(n-l)I U ; ' 

may be found by starting with Ph instead of Ph n in (2). 



226 INFINITESIMAL CALCULUS. [Ch. XXXIX. 

B n is the amount of the error when the first n terms of 

the series are taken as the value of f(a + h). Thus for 

log (a+h) (§206) the numerical value of the error would 

h n h n 

lie between — - and — j-r-, the greatest and least values 

na n n(a+h) n 

of R n . 

The method of making small corrections explained in 

Ch. XI is equivalent to the use of the first two terms of 

Taylor's Theorem. In this case the error is therefore 

%f" (a+ dh)h 2 , where O<0<1. 

209. Maclaurin's Theorem. Taking a=0 in (5) and 
writing x for h, we have 

/(*)=/(0)+/'(0)z+^W. . .+Rn, (7) 

where R n J^^l^ or =ffiM(i-^)n-i^ 

according as Lagrange's or Cauchy's form is adopted for the 
remainder. (7) is the statement of Maclaurin's Theorem. 
The expansion is therefore possible if f(x) and its first n 
derivatives are continuous from x=0 up to the value of x 
adopted in the series. 

210. Maclaurin's Series. Taylor's Series. If j(x) can be 
expanded by means of the series (7), and if the values of x 
are such that £ n _oo R n =0, then 

or f(x) is equal to the limit of the sum of the terms of an 
infinite power series (Maclaurin's Series). 
Under similar conditions we obtain from (5) 

Ka+h)=f(a)+f(a)h+ f -^h2 + 

which is Taylor's Series, 



209, 210.] TAYLOR'S THEOREM. 227 



x 3 ar 5 



Ex. 1 . From (7) sin x =x — — + — — . . . + Rn, where Rn (Lagrange's 

o! 5! 



(7T \ X X XXX 

6x + n—)—-. Now — r — — . — . — . . . , and each f rac- 
2/ n\ n\ 1 2 3 

tion after a certain point is numerically <1, hence the limit of the 

product = for n = co . Also sin ( 6x + n— ) remains finite as n 

increases, since it cannot be >1 or <— 1. Thus £ n =oo Rn = 
for all values of x. Hence, for all values of x, 



X s x* 
m*z-x-f+ t 



X 2 x 4 



Similarly, cos x = 1 — — +— : — . . . for all values of x. 



/y* it /v»0 



2. log (l+x)=x— - + — — . . ,+Rn, where Rn (Cauchy's form) 



( - I)"" 1 (1 - d)n-^ ( - I)"" 1 /l - d 




n 



(1 + Ox)* 

But l-0|<|- + if -1<z<1. Also 1-0 remains finite. 

x ~~ 



Hence, for these values of x, 



x 2 x 3 



£n=ooRn = 0, and log (l+x)=x- — + , 

For all other values of x the series is non-convergent (§196) and 
hence cannot represent the function. 

3. Show that e x = l +x + —~ +—+... for all values of x. 

Z 1 o ! 



CHAPTER XL. 
FOURIER'S SERIES. 

211. (a) A function f(x) may be developed into an infinite 
series of the form 

A +ai cos x +a 2 cos 2x + . . . +a n cos nx + . . . , (1) 

consisting of a constant term A, and cosines of x and mul- 
tiples of x, with constant coefficients a l7 a 2 , . . . For any 
value of x from x=0 to x=n the series will represent the 
function, i.e., the limit of the sum of the terms of the series 
will be equal to the value of the function. 

(b) f(x) may also be developed into a sine series of the 
form 

a\ sin x + a 2 sin 2x+ ... +a n sin ?ix+ ... (2) 

and the series will represent the function for any value of x 
between and n. 

(c) f(x) may be developed into a sine and cosine series 
of the form 

A+ai sin x+a 2 sin 2x + . . .+&i cos x-\-b 2 cos 2x + . . . , (3) 

and the series will represent the function for any value of x 
between —tz and n. 

Whether the series represent the function for values of x 
other than those stated will depend upon the nature of the 
• function. 

228 



211-213.] 



FOURIER'S SERIES. 



229 



The above propositions were fully investigated for the 
first time by Fourier (Theorie analytique de la Chaleur, 1882) 
and the series are called Fourier's Series. 

212. Assuming the form of the series we shall explain a 
method of calculating the constants. It will be necessary 
to make use of the following results of integration which may 
be easily verified by the student (see Ch. XXII). 

Supposing n and m to be integers and n^m, the follow- 
ing integrals have the values herewith given: 



I cos nx dx 

/ sin nx dx 

/ cos mx cos nx dx. 
j sin mx sin nx dx . 
I sin mx cos nx dx. 

I cos 2 nx dx 

I sin 2 nx dx 



From to n. 


From — n to tc. 































4* 



i« 



It 



7Z 



213. The cosine series. Suppose 

f(x) =A+di cos x+a 2 cos 2x + . . .+a n cos nx + . . . (1) 

An operation may be performed which causes every term 

of the series to disappear except that of which the constant 

is desired. Multiply (1) by das and integrate between and n. 

Then 

1 r* 



1; 



j{x)dx=An, .*. A=— j(x) dx. 
^ j 

Multiply (1) by cos nx dx and integrate between and n. 
Then 



i: 



f(x) cos nx dx=a n . ^7r, 



<x n — 



2 

7Tj 



r« 







f(x) cos nx dx. 



230 



INFINITESIMAL CALCULUS. 



[Ch. XL, 



Ex. 1. Let f(x) = x. Then A = 



1 



71 



Also, a n = 



- 



~ 



*As KX/Jj p. . 



x cos nx dx= »(1 "~cos nn). 



7171' 



Hence, making n = 1, 2, 3, . . . , 



x 



* 4 / 1 1 \ 

= — — (COSX + -— cos 3x + — COS DX + . . .1 
2 71 \ 3 2 5' I 



In Fig. 124, I, II, III represent the graphs of the first three terms 

,0 from x = to x = tz 1 and AB that 
of their sum. The limit of AB 
for the infinite series is the straight 
line OC, or y = x. 

The series holds from x = to 
x = 7i ; for smaller or greater values 
of x the series does not represent 
x. The value of each term of 
the series is unchanged when the 
sign of x is changed, and is re- 
peated whenever x changes by the 
amount 2n. Hence the graph of 
the series consists of the lines of 
Fig. 125 continued indefinitely in 
both directions, or the equation 
of all these lines is 




Fig. 124. 



71 



*--2 



1 COS X 

71 



1 



cos 3x + 



...). 




Fig. 125. 



The axes may be transferred to any other position in the usual 
way. Thus to make the middle point of OA the origin, the ar-axis 



213.] FOURIER'S SERIES. 231 

being parallel to that of the figure, change y into y + in, and x 
into x + hx. The result is 

4 / . 1 . o \ 

y = — sin x — — sm 3# + . . . J . 

U 7Z \ 3 2 / 

mn 4m / 1 \ 

Since m^=— I cos x + — cos 3z + . . . ) , the terms on the 

2 7T \ 3 2 / 

right represent any line y = mx through the origin, x varying from 
to 7r, and the equation of all lines like those of Fig. 125 making 
angles ±tan _1 m with the x-axis is 

mn 4m / 1 \ 

y = - 7r (cos£ + -— cos 3a; + . . .) . 

2 7z \ 3 2 / 



111 7T 



2 

and 



2. Making x = in Ex 1, show that — +— + -- + .. . = — , 

1 o O o 

1111 7T 2 ,1111 K* 

hence that — +—+—+— + ... = —, and — -—+ — -— + ... = ^. 

2/1 1 

3. x 2 = — — 4 (cos a; — — cos2a; + — cos 3x — . . .] , for [— n, rc].* 

In this, as in all other cases, the cosine series of an even func- 
tion (see Ex. 12, p. 40) represents the func- 
tion for negative values of x as well as 
for positive values. The graph consists of 
a series of parabolas of breadth 2n and -j qt 

height 7i 2 (Fig. 126). Fig. 126. 

2 4 /cos 2x cos 4a; \ 

4. sm x = ( - — r- + -r — — +...), for [0, 7rj. 

^ 7r \ 1 . 3 3.5 / 




sinh n 
5. cosh # = 



71 

hence 



"-, /cos£ cosza; \ "1 

_ X ~ 2 VI+P"!^ 7 + * ' 7 J' for [ -*' ^ and 



111 w 

+ 1 t 02 + -, , Q2 + . . . = i(7T COth 7T — 1), 



l + l 2 1+2 2 1+3 
1 1 1 



l + l 2 1 + 2 2 1+3 

* See § 22. 



— . . . = J(l — 7i cosech tt). 



232 



INFINITESIMAL CALCULUS. 



[Ch. XL. 



2 14. The sine series. Each of the cosine expansions gives 
on differentiation a sine expansion of the form 

f(x) =ai sin x+a 2 sin 2x+ . . . +a n sin nx+ . . . (1) 

The series may be obtained without reference to the cosine 
series as follows: Multiply (1) by sin nx dx and integrate 
between and n. Then 



f(x) sin nx dx=a n . \tCj .'. a n =— 

71. 



fix) sin nx dx. 



4 
Ex. 1. Let/(x) = l. Here a n = — (1— cos W7r). 

nn 

4 . 

Hence 1=— (sinx+J- sin Sx+i sin 5x + . . . ). 

Fig. 127 represents the graphs of the first three terms and of 
their sum. The limit of the latter graph is O'C, the line y = l 
The limit of the sum of the series is 1 for all values of x between 
and 7r. Thus the limit of the y of the graph of the series for 
2 = or it from inside the interval is 1, but its value for # = 0or 
n is 0. The series 

— (sin x+f sin 6X -f . . .) 

represents any constant h, the amplitudes of the sine curves 
being 4h/7t, 4h/2>n, . . . The complete graph of the series con- 




^ 



h\ 



-7T 



7T 



Fig. 128. 

sists of the straight lines of Fig. 128 continued indefinitely in 
both directions, and the equation of all these lines is 

j/ = — (smz+f sin 3x + . . . ). 

K 



214 125.] 



FOURIER'S SERIES. 



233 



Although the series is convergent, the series formed by the 
derivatives of its terms is non-convergent, and therefore does 
not represent the slope of the graph (the derivative of the func- 
tion) at any point. 

2. # = 2(sin x — \ sin 2x+-$ sin 3x — . . . ). 

This represents x for 0<x<tc. Since the series changes sign 
with x y and the function is an odd 
one, the series represents the func- 
tion for negative as well as positive 
values of x. Hence it holds for 
— n<x<n. The graph consists of a 
series of straight lines, as in Fig. 129. 




Fig. 129. 



3. cos x = — 



4 /2 sin 2x 4 sin 4a: 



/A sin ax 4 sin 4x \ _ _^ _ 

Draw the graph. 

, 2 r in 1 4 \ . /7T 2 4 \ . n 

4 - x A\r-v) smx+ (j-¥) smSx+ --- 



7T 2 . 7T 2 "1 

— — sin 2x— — sin 4x — . . . , for [0, 7r[. 



5. From Ex. 4 show that — — ^+^ 

loo 

6. Show that 



7T 



32 



— = sin a; +i sin 3a: + J sin 5a: + . . . , for ]0, n[, 

•— = cos x — J cos 3a: +i cos 5a: — . .. , for ~~ o"> n"! • 



215. The function represented by a Fourier series need 
not be a single function throughout the range iz of the value 
of x; the same series may represent one function for a part 
of the range and one or more other functions for the remainder 
of the range. 

Let }i(x) =A+di cos x + a 2 cos 2a: + . . . (1) 

for x=0 to x=a, and 



f 2 (x) =A+ai cos x+a 2 cos 2z + . . . 



(2) 



234 



INFINITESIMAL CALCULUS. 



[Oi XL. 



(the same series) for x=a to x=n. Multiply (1) by dx 
and integrate between and a, also multiply (2) by dx 
and integrate between a and n, and add the results. Then 
each term of the series is, on the whole, integrated between 
and n* Hence 



/i (x) dx + 



f 2 (x) dx=Anj 



.-. A = 



=—\ fi( x ) d> x + h( x ) d x V 

^ L J J a J 

2rf a f* ~i 

Similarly, a n =— /i (x) cos nz dx + /2OE) cos nx dx 
x*-J J a J 

for the cosine series, and 

2 r f a ■ f* "I 

a n =— /1 (z) sin nx dx + / 2 (x) sin nx dx 

ttLJo J a J 

for the sine series. 

The series may not hold at the point or points where 
the change of function occurs. 

It may be noticed that A in the cosine series is always 
equal to the mean height of the graph from x=0 to x=n. 



-7T . f 



¥ 

Fig. 130.* 



7T 



Ex. 1. To find a cosine series which =1 for 0<x<%n, and 
for \it<x<Tt. 



= 



A=K> an d An 



71 

2p 

71 
J 



7T 

2* 2 . 717T 

cos nx ax = — sin ~pr . 

nn 2 



Hence the series is 



1 2 

— +— (cos x — \ cos 3a;+j cos 5x — . . . ). 

2 71 



* The electrician's make -arid-break curve. 



216,217.] FOURIER'S SERIES. 235 

It is true when x = and x = tt, but = ^ when x = ^tu. 
2. Find a sine series for the same. 

2 /sin x 2 sin 2x sin 3x sin 5x 2 sin 6x sin 7x \ 

1 2 

= 2 +— (sin 2x+i sin 6#+i sin lOx + . . . ). 



Arts, 



(See § 214, Ex. 6.) 

216. The cosine and sine series. For values of x between 

— 7z and 7i j 

fix) =A +ai cos x + . . . +a n cos nx + > . . +6i sin x + . . . 

+ 6 n sin n# + . . . 

It is easily shown, as in § 213, that the constants may 
be determined as follows: 



*=k 



fix) dx, a n =— fix) cos nx dx } 



71 



71 



J —7C 



71 



f{x) sin nx dx. 



2 sinh 71 /l 



_ z smn n /i cos W7r 

Ex. e* = ■ ( — + . . . H — - — - cos nx + . . . 

7T \2 n 1J rl 

U COS 727T . \ 

— — — — — - sin nx — . . . J . 

7i 2 + l / 



217. By the following method a cosine series which will 
hold for values of x from to any number c (instead of n) 
may be obtained. If x=cz/tz, x=0 when z=0, and x=c 
when z=7i. Hence, in fix) change x into cz/tz, develop in 
terms of z, and change z into tzx/c. Similarly to obtain a 
sine series for values of x between and c, or a cosine and 
sine series for values of x between — c and c. In all cases 
the constant term is equal to the mean height of the graph 
(or the mean value of the function) for the interval in ques- 



236 INFINITESIMAL CALCULUS. [Ch. XL. 

tion. In this way the series already obtained may be adapted 
to the intervals stated below. 



_ C 4C / TlX 1 6t:X \ r ^ _ 

Ex. 1. x =77-- i( c os — +— cos + . . .) , [0, c]. 

A 7Z \ Co C ' 

, c 2 4c 2 / 7tx 1 2nx \ _ 

- _ 4/z, / . 7r;r 1 . 37T£ \ 

3. ft = — sin h— sin + . . .) , 10, c[. 

7z \ c 3 c 1 

2c [ . TZX 1 . 27:2 V 

4. x =- sm — --sin K . .) , J — c, c[. 

7i \ c 2 c / 

7T . 7T:T 1 . 37TX _ 

5. — = sin — + — sm h. . . , JO, c[. 

4 c 3 c 

* 
218. If a function of z is developed into a series for the 

interval — c to c, and if the values of the function are repeated 
periodically for every interval 2c of x, the series will con- 
tinue to represent those values as x increases or decreases. 
In other words, the periodic function of period 2c is developed 
into a series consisting of a constant term and harmonic 
functions of periods 2c, 2c/2, 2c/3, etc. Fourier's Theorem 
is to the effect that this development is always possible, 
the complete series being of the form 

TZX 2tzx , , , . TZX . 2tzx 

A+ai cos ±-a 2 cos h . . . +&i sm — + & 2 sm K . . • 

c c c c 

which is equivalent to 

(TZX \ (2tzx \ 
Voi\\ +A 2 sin( \-a 2 ) +. . . , 

where A n =Va n 2 +b n 2 and a w =tan"" 1 (o n /6 n ). 

As already stated, the function may consist of distinct 
functions for parts of the interval. 



218.] 



FOURIER'S SERIES. 



237 



Examples. 
Develop the functions represented by the following figures : 




A 

1. Fig. A. 

2. Fig. B. 

3. Fig. C. 




5. Fig. E. 

6. Fig. F. 



h 4ft ( 
--—(cos— + 

2 TZ 2 \ C 

8ft t . Ttx 1 . Stzx 

_( sm ___ sm _ + 

2ft / . tzx 1 . 2nx 

— sm — sin h 

tz \ c 2 c 



...). 
...). 




h 



-c 



E 

ft h I . tzx 1 . 

— (sm h— si 

2 tz \ c 2 



F 

2tt# 

sm f- 



tzx 1 

4- 

2 



c 

27T^ 



+ — (sm h— sm 

2 tz \ c 



...). 
...). 



4ft / . 7TX 1 . 37nr 

— ( sm h— sm h 

TZ \ c 3 c 



• • . i • 



ft 



-*c 



G 



7. Fig. G. 

8. Fig. EL 




-c 




ft 



c 2c 




H 



A 2ft / . tzx 1 . Stzx \ 

- (sm- f-— sin - K . .) 



2 7T \ 

ft 2ft/ 

2 (cos— + 

4 7T 2 \ CO 2 



c 3 

TZX 

c 



c 
Stzx 



6TZX \ 

- +...). 

C / 

ft / . 7nc 1 . 2tzx \ 

-\ — (sm — sm K . .) . 

TZ \ c 2 c / 



238 INFINITESIMAL CALCULUS. [Ch. XL. 



9. Fig. J. (Parabolas. Latus rectum = 2c). 

1 2nx \ 

+ cos -- + ...). 



c 2c I tzx 

3-A C0S 



10. The displacements of a slide-valve actuated by a Gooch 
link were measured at eight intervals each of 45°, and found to 
be as follows, beginning with the crank on the inner dead-centre: 

2*44, 1'65 ; 0, -T37, -L87, -T37, 0, T65. 

Assuming that the motion of the valve is compounded of 
two simple harmonic motions, one of double the frequency of 
the other, as represented by the equation 

y = k+a sin (0 + a) +6 sin (20+/?), 

where is the crank angle, find the values of k, a, a, b, /?. (Castle, 
Manual of Practical Mathematics.) 

There are various graphical or other practical methods by 
which the coefficients of a small number of terms of a Fourier 
series may be found, but in this example an algebraical solution 
will suffice. Assume 

y = k+a 1 sin J r b l cos d +a 2 sin 26 +b 2 cos 2d 7 

substitute the given values of y for = 0, 45°, 90°, etc., and solve 
the equations. 

Arts. 2/ = *14+2 , 16 cos 0+*14 cos 20, 
or ='14-2-16 sin (0+90°) +'14 sin (20+90°). 



I 



CHAPTER XLI. 

APPROXIMATE INTEGRATION. ELLIPTIC INTEGRALS. 

219. Approximate integration. If the general value of 
f(x) dx cannot be obtained it may be possible to find a 



sufficiently close approximation to the desired result. 

(1) If f(x) can be developed into a rapidly converging 
series, the integration of a few terms will give an approxi- 
mate value of the integral. 

(2) The curve y=f(x). may be plotted when f(x) is given. 
Its area obtained by Simpson's Rule (§ 131) or by the pla- 
nimeter (Appendix, Note D) will give an approximate value 



of 



f(x) dx between assigned values of x. 



(3) 



y 2 dx and y 3 dx as well as \y dx for a curve which 

has been drawn mechanically or otherwise can be obtained 
mechanically. The result, although theoretically exact, is 
affected by observation and instrumental error. On Mechan- 
ical Integration see Appendix, Note D. 

Elliptic Integrals. 

f dd , |Vl-m 2 sin 2 # dd, 

Jvi — ra 2 sin 2 



220. 



dd 
and 



1 



(1+a sin^Vl — ra 2 sin 2 # 

239 



240 INFINITESIMAL CALCULUS. [Ch. XLT. 

are called elliptic integrals of the first, second, and third 
class respectively. The constant m, which is assumed to be 
not greater than unity, is called the modulus of the integrals. 
The lower limit is understood to be in each case, and, the 
angle varying from to 6, 6 is called the amplitude of the 
integral. The integrals are represented by the symbols 
F(m, 6), E(m, 6), and II "(a, m, 6), respectively; orby.F m (#), 
etc. When the limits are and \tz (i.e., when the ampli- 
tude is \iz) the integrals are said to be complete. 
If sin 0=x, the integrals become 



I 



dx 



and 



V (1 — x 2 ) (1 — m 2 x 2 ) 

dx 



l — m 2 x 2 , 
ax, 



x 2 



J (l+ax 2 )V(l-x 2 )(l-m 2 x 2 )' 



and they are complete when the limits are and 1. 

221. The values of the elliptic integrals cannot be expressed 
in finite terms, but may be approximated to by infinite 
series. • 

Thus by the Binomial Theorem 

= (l-m 2 sin 2 fl)-*d0 



Vl — m 2 sin 2 # 



= (l +-m 2 sin 2 + y~i^ sin 4 # +■ ' ' fi m 6 sin 6 fl + . . .) dd, 

and each term may be integrated by § 113 (see Ex. 12 below). 
Taking the limits as and \n we have (§ 120) for the 
complete elliptic integral of the first class 

•nt ix rcf"i , /l \ 2 /1.3 9 \ 2 /1.3.5 a 2 i 



221,222.] ELLIPTIC INTEGRALS. 241 

Similarly for the integral of the second class we have 
Vl-m 2 sin 2 0d0 = (l-?n 2 sin 2 0)*d0 

= (l- \m? sin 2 0- ^—rfi sin 4 0- * : 3 - m 6 sin 6 0- . . .) dd, 
\ 2 2.4 2.4.6 / 

and 

ttt /l \ 2 1/13 \ 2 1/135 \ 2 T 

sL^W -sfc™ 2 ) -5(2^4-6^) "•••J 

for the complete integral of the second class, E{m,\n). 

Three-figure tables of the integrals for certain values of the 
modulus and amplitude are given at the end of this volume. 

It may be noticed that 



also, E(l, 6) = 



JS(0, 0) ==F(0, 0) =0 (in radians); 
cos d0=sin 0, 

' dd , . /7T 



cos 



=log tan (I +|)=;«?). 



222. From the above expansions and the integral (§ 113) 
of sin n dd it may be shown that 

E(m, nn±6)=2nE±E{m J 0), 
F(m, nn±d)=2nK±F(m, 0), 

i? and if being the values of the integrals for the amplitude 
%tc, and n being any integer. Hence a table of the elliptic 
integrals in which the amplitude varies from to \n may 
be used for all higher values of the amplitude. 

Examples. 

x 2 xi 2 
1. To find the length of an arc of the ellipse — + 7 -- = l. 

a 2 o 2 

The complement of the eccentric angle being denoted by 6 we 

have x = a sin 6, and y = b cos 6. 

: . dx = a cos 6 dd, dy= —b sin 6 dd 



"■ 



242 



INFINITESIMAL CALCULUS. 



[Ch. XLI. 



whence ds 2 = dx 2 +dy 2 = (a 2 cos 2 6 +b 2 sin 2 6) dd 2 

= [a 2 - (a 2 - b 2 ) sin 2 6>] dd 2 = a 2 (l -m 2 sin 2 /?) dd 2 , 

where m fc is the eccentricity of the ellipse. Hence the length of 
the elliptic arc measured from the end of the minor axis is 



a 



Vl -m 2 sin 2 dd = aE(m, X ), 



an elliptic integral of the second class. The length of the quad- 
rant of the ellipse = aE(m, %n). 

2. Find the circumference of the ellipse x 2 +2y 2 =2. 

Arts. 7' 64. 

3. Of the ellipse 3x 2 +±y 2 = 12 find (1) the length of the arc 
from x^O to x = l, (2) the length of the quadrant, (3) the middle 
point of the quadrant. Arts. T036, 2*934, (1*36, T27). 

4. An arc of the lemniscate r 2 =a 2 cos 26. 
From ds 2 =r 2 d0 2 +dr 2 we have 



-J: 



dd 



Vl-2sin 2 



Let 2sin 2 #=sin 2 0. Then 

'fa d(f> 



a 

V2 



=4^ 



o Vl-i s in 2 V2 \V2 



■> 0i), 



an elliptic integral of the first class. The length of a quadrant 
of the lemniscate is therefore 



a 



F 



(■ 



i^i — 



V2 \V2 
If 6=30°, show that s=*584a. 



} 2 



n) = l'311a. 



Cc 



5. 

6. 

7. 



dx 



=—F 



a 



, sin 



— i 



V( a 2 -x 2 )(b 2 --x 2 ) a 

^ dx 1 /vV-6 2 t c 

, =—F( , tan^r 

V(a 2 +x 2 )(b 2 +x 2 ) a \ a b 



Let x = b sin 0. 



dx 



F 



b 



V(a 2 +x 2 )(b 2 -x*) V a 2 + b 2 Wa 2 + b 2 



>> co *- l y- 



222. ELLIPTIC INTEGRALS 243 



. I ™ =2F(m, sin- 1 ^). 



8. I -y- 

V x{l —x)(l —m 2 x) 

9 A simple pendulum of length I oscillates through an angle /5 
on each side of the vertical. To find the time of an oscillation. 

When the pendulum makes an angle (f> with the vertical, the 
acceleration — g sin $ in the direction of the motion = d 2 s/dt 2 = 

I d 2 <t>/dl\ 

d 2 <f> g . '> 

Multiply by 2d</> and integrate. Then 

(S) =f( C0S ^~ C0S ^ = f( sin2 ^- sin H^). 
Hence solving for d£ and integrating, 



2\|<7 



w d4> 

(1) 



o Vsin 2 •§•/? — sin 2 J 
is the time of a half oscillation. Let sin £0='sin i/? sin 0. Then 
(1) becomes 

X \ J - f , *" = x &( S in tf, W . 

\<7Jo Vl-sinH/Ssin 2 ^ \9 ' 

Hence the time of an oscillation is 

T 



4 



^(sin ift **). 



10. Find the time of oscillation of a pendulum when a =60°. 

Arts. 3'372V77<7. 
Find the time through the lower half of the motion. 

Ans. 2^-i^sin 30°, sin-^^j =V102Vl/g. 

11. If the arc s is small compared with the length I, show that 
the time of oscillation of a simple pendulum is approximately 



^K l+ 6il)- 



9 

12. Show that 

F(m, 0) = 0+im 2 (0-sin cos 0) 

+ <ft-m 4 (30-3 sin cos 0-2 sin 3 cos 0)+. . ., 
E(m, 0) = 0-im 2 (0-sin cos 0) 

-Am 4 (30-3 sin 0cos 0-2 sin 3 cos 0)-. . . 



CHAPTER XLIL 
SINGULAR FORMS. 

223. We have already seen that for a certain value of 
the variable a function may assume the form 0/0. The form 
is said to be singular; it is also called an indeterminate form. 

There are other singular or indeterminate forms, such 
as 00 /oo , . 00 , 00 -00 , 0°, 00 °, l 00 . 

A function in a singular form has no value c Our object 
is to find the limit of the value of the function as the vari- 
able approaches the critical value in question. 

224. The form 0/0. The fraction (x— l)/(z 3 — 1) takes 
the form 0/0 when x = l. But 

£—1 x—1 1 



x s - 1 (x-l)(x 2 +x + l) x 2 +x + l' 

provided that x— 1^0. The last fraction —^ when x = \, 
hence the given fraction =J when x=l. 

Method of the Calculus. Let the fraction be J{x)fF{x), 
and suppose a to be the value of x which causes the frac- 
tion to take the form 0/0, or that /(a) =0 and F(a) =0; also 
that x=a+h, where h is a small quantity which is to =0. 
Then, assuming the functions to be such that the expansion 
of Taylor's Theorem applies, 

f(a+h) /(«)+/W+r^ 2 +— f'(a)+ f ^-h + ... 

v T ' F(a)+F'(a)h + Y A 2 L h 2 + ... F'(a) +-y-y^+- ' ' 

244 



223-225.] 



SINGULAR FORMS. 



245 



Hence when h^O, i.e., when x = a, the given fraction 
if (a)/*" (a), or 

f(x) f'(a) 



£ 



F(x) F'(aY 



If /'(a) and F'(a) are also 0, it may be shown in the same 

way that £^- =p77JTy and so on. 

__ , _. /(*) x-1 f'(x) 11, , 

Ex.1. If -=7-.= — ; — T> "7^r"x = ^~, = TT w henx = l. 
.F(x) x 3 -l JP'(x) 3x 2 3 

x — 1 1 

The work may be conveniently expressed thus : When iil, 

x-l 1 



*V-1 3x 2 Ji 



1_ 

3 



># c— x pX ±0— x 



pX _|_ />- 

2. If z = 0, £ . " =— — 

sin x cos a; 

/Or) e*-l-log'(l+x) 



3. 



TO 



X' 







when a: = 0. 



fix) 

F'(x) 



>z 



1 



■=— when # = 0. 



2x 







/"(*) 
F"(x) 



(l+x) ; 



*= 1 when x = 0. 



'. £ ° = 1 when # = 0. 



X' 



225. The form 00/00. Let / (x)/F (x) =00/00 , first when 
£ = 00. Let the graphs of f(x) and jP(x) be PQ and P'Q', 
Fig. 131, and let OM=x. Let the limits of the tangents at 



246 



INFINITESIMAL CALCULUS. 



[Ch. XLn. 



P and P' be the asymptotes AS and A'S' when x = <x> , and 
let A'A=c. Then MP=}(x), MP' =F(x), tan MTP=f'(x), 




O A' 



A T M 

Fig. 131. 



tan MT'P'=F'(x). Hence 

f(x) T Mfjx) 
F{x)~T'MF'{x)' 



But 



.TM 



AM 



£>T'M *A'M 



& V A' Mi 



when A'M = oo . Hence 



,m nx) 

A: 77 /~A ^o 77f/ 



X 



(i) 



Secondly, let f(x)/F(x) =oc/oo when x = a. For s sub- 
stitute a + l/z. Then 2 = 00 when x = a. But by (1), if 

2 = 00, 

', ( . + i)-^ { „ + i)( -i) _ Vi)' 



or 



. /(*) x /'w 



226,] SINGULAR FORMS. 247 

Hence the result (1) holds in this case also. Thus when 
a fraction has the form oo /oo the limit of its value is found 
from the same differentiations as when it has the form 0/0. 

. TZX 

,, s log cos — 

Ex. M = -V4 = -when^l. 

r (x) log(l— x) oo 

71 TZX 

f(x) ~2" tan ^ n 1-x \ 

7^t-t = = — . = — when x = 1. 

F'(x) 1 2 nx 

cot 



x tz „ 1 —x tz — 1 

But (§224) -£ 



l-x 2 

= 1 



2 ^x 2 7i 7ro: 

cot — -- cosec 2 -- 



Hence the given fraction = 1 when x = 1 . 

226. The forms 0.00 , 00 -00 . A function which assumes 
the form O.00 , or 00-00 may, by an algebraical or other 
change, be made to take the form 0/0 or 00 /oo . 



a 



Ex. 1. x(l —e x ) tends to 00 . when x = 00 . The limit is most 
easily found by using the exponential series. 



a 

For 



Jf( i-.-: ) .-,[i-(i-i + £-...)] 



a 2 



= a— — +. . . = a when # = oo . 
2x 

TZX 

2. (1 — x) tan — tends to the form . 00 when x = 1. 

1 —x . 
But it = , which = — when x = 1. 

TZX 

cot- 

— 1 2 

.'. (§ 224) we have ==— when x = l. 

TZ TZX TZ 

--cosec'- 



248 INFINITESIMAL CALCULUS. [Ch. XLII 

.\ £{l —x) tan — =— when x <~ 1. 

D 

a; 1 

3. — -— , = oo—oo when x =1. 

x — 1 log x 

_ J a: 1 a: log a:— x + 1 . 

But r — ; = — ; tti = — wnena;=l, 

# — 1 log a: (a; — 1) log a; 

lo<r x 

whence (§ 224) z , which = — when x = l. 

1 hloga; 

x 



. . . x 1 

A second differentiation gives ~- -, which = — when x = l. 

X 2 X 

. . x 1 

.*. i is the limit of - — ; when x=l. 

x — 1 log X 

227. The forms o°, 00 °, i 00 . Functions which assume 
these forms may be made to assume the form . 00 and 
therefore 0/0 or 00 / 00 by first taking logarithms. 



Ex. 1. f(x) = x l0 * sinx tends to the form 0° when x = 0. 

_ . r/ N a _ log x 00 

But log fix) = . ; — . log x = a. ; — = — when x « 0. 

log sin x log sin x °° 

1 

Differentiating (§ 225), a = a = a when x ^ 0. 

cos a; x 

sin x 

-*. £ log /(a?) -a. But * £ log /(a?) -log £/(*), .\ £f(z) = e*. 

1 
2. f(x) = x 1 ~ x tends to the form l 00 when #=1. 

* If v is any variable and £v = b, (&H0), then (§8) v = 6+t= 

m+i/b). 

.\ log v — log&=log (l + i/b), whichji.0. 
■. ^l g v=log6 = log^. 



227.] 



SINGULAR FORMS. 



249 



_ i ,, x 1 i log x . 

But log f{x) = ~ — - log x = : = — when x = 1. 



1-x 



1-x 



/. £log/(^)=- 1 



_i 



-i- 1 

= — 1, .\ x l x ~— when x= 1. 

e 



Examples. 
1. When # ^ show that 

(l)-S^l, (2)^5^1, 



(4) 



a: X 

cos a; — cos mx 1 — m 2 



cos a: — cos wx 1— n 



2 > 



log sin 2x 

(o) ■; : = 1, 

log sin x 



a + x 

(8) a^ = 



e a , 



(3) 



->%■ — o — X 



= 2, 



log (1 +x) 

2 ? 

(7) a^loga; = 0, 



/r\ l°g SeC X 1 

( 5 ) 2 ' 



(9) x x = 1. 



s 3 -3a;+2 
2. — — — — - = when x =\. 



x 3 +4x 2 -5 



3. Es = oo, (1) — = oo, (2) 2* sin ^ = a. 

4. Ha? = |-, (1) (sin a;) sec2 * =;—=., (2) sec a; (|-a; sin aA = 1. 

5. (sin x) tan *= 1 when a; =0 or -. 

1 +— ) = 6 a when a: = oo , and = 1 when x = 0. 

„ sec a; cos 3x n . 7r 

7. — - = = — 3 when x = — . 

sec 3# cos re 2 

^ tan a; cos 3a: sin x , ^ 

8. ; - = ■ ^-—-(-3)(-l) = 3 whena:~~ e 

tan 3a: cos a; sm 3x 2 



9. 



(e*-l) tan 2 x /e* — 1\ /tan z\ 2 



a;* 



- £r) (^) - 



1 when x = 0. 



CHAPTER XLIII. 

SUCCESSIVE DIFFERENTIALS OF FUNCTIONS OF MORE 
THAN ONE VARIABLE. EXTENSION OF TAYLOR'S 
THEOREM. MAXIMA AND MINIMA FROM TAYLOR'S 
THEOREM. 

Successive Partial Differentials. 

228. Suppose u to be ax 3 —xy 2 +y. We have as in § 45, 
supposing x alone to vary, 

d x u = (3ax 2 —y 2 )dx, d x 2 u=6ax dx 2 , d x 3 u=6adx 3 , d x 4 u=0, 

d y u = (— 2xy + l)dy, d y 2 u = — 2x dy 2 , d y 3 u =0. 

Again, d x u or (3ax 2 —y 2 ) dx contains y as well as x, and we 
may obtain its differential on the supposition that y alone 
varies. We then have 

d v d x u = — 2ydy dx, d v 2 d x u = — 2 dy 2 dx, d v 3 d x u =0. 

Similarly, d x d v u = — 2ydx dy, d x d v 2 u = — 2dx dy 2 , d x d y 3 u =0. 

229. In comparing these results it will be seen that 

d x d y u=dyd x u, d x dy 2 u=d y 2 d x u, d x d y 3 u =d y 3 d x u; 

also, d x dyd x u=d x 2 d y u=dyd x 2 u; in other words, the succes- 
sive operations indicated by d x and d v may take place in 
any order. 

It will be shown that this is true generally. 

230. Continuity of a function of two variables. Let 
u=j(x, y), and let dx and dy be infinitesimal increments of x 

250 



228-231.] SUCCESSIVE PARTIAL DIFFERENTIALS, 
and y. Then f(x, y) is continuous at x, y, if 



251 



£f(x+dx, y + dy) =f(x, y), 

when dx and dy approach the limit zero in any manner 
whatever. 

If in Fig. 132 OA=x, AB=y, AD or BG=dx, BH=dy, 
and BP=f(x, y), then EQ=f(x+dx, y+dy). The condition 




of continuity implies that £EQ = BP when E is any point 
near B in the plane XOY. 

In what follows it is assumed that the functions and 
their derivatives are continuous for the values of the variables 
under consideration. 

231. Let A x indicate an increment produced by the incre- 
ment dx of x, y being regarded as constant, A y having a 
corresponding meaning. Then if u=f(x, y), A x A y u=A y A x u. 

For, A y u=J(x, y+dy)-f(x, y), 

A x A y u=f(x+dx, y+dy) — f(x + dx, y) 

-[/Or, y+dy)- f{x, y)] 

= }(x+dx ) y+dy)-f(x+dx, y)-f(x ) y+dy)+f(x, y). (1) 



252 INFINITESIMAL CALCULUS, [Ch. XLIII. 

The symmetry of the result shows that it would also be 
obtained for A y A x u, 

Ex. In Fig. 132 let u be the volume COAB . P. Then (1) 
expresses that 

HBGE . Q = FODE .Q-CODG . I-FOAH .J -{-COAB . P, 

which is obvious from the figure, as is also the fact that 
HBGE . Q is A y A x u as well as A x A yU . 

232. Since u and 'its derivatives are assumed to be con- 
tinuous at and near x, y, 

JyU=dyU+I 2 

(§ 42), where I 2 is an infinitesimal of at least the second 
order, and 

A X AyU =A X {d y u + 1 2 ) = d x dyU + h J 

where ^3 is of least the third order. Hence 

d X dyU AxAyU 

dx dy dx dy ' 
Similarly ' ^=4^ ^^ x u=A x A v n. 

Hence, dxdy 2 u=dxdydyU=dydxdyU=dyd y dxU=dy 2 d x u J and 
similarly for any combination. 

These results may obviously be extended to functions of 
any number of variables. 

_. . d x 2 u d y 2 u d x dyU dyd x u d x 3 dyU 

The expressions -t-«-, -3-9, 1 — t-, 3 — ir> 7 o -, ? etc. are 
^ aaH a?/ J ax dy dy dx dx 6 dy 



frequently written 

d 2 u d 2 u d 2 u d 2 u d 4 u 



dx 2 ' dy 2 ' dxdy' dydx' dx s dy 



, etc. 



232, 233.] SUCCESSIVE TOTAL DIFFERENTIALS. 



253 



Ex. I. In Fig. 132, u being as before the volume of OP, 

dxdyu _ J x Ayu H BGE . Q 
. dxdy~^dxdy~^ HBGE' 

i.e., the limit of the mean height of the solid BQ, which limit 



is BP or z. Hence d x d y u = z dx dy, .'. the volume = 
tween assigned limits, as in § 179. 



z dx dy be- 



2. Verify that d y d x u = dxdyU or 



d 2 u 



d 2 u 



dy dx dx dy 
(2) u = sinxy, (3) u = ta,n~ 1 (y/x). 

3. If u= (2x — 3y) s , verify that d x dy 2 u = dy 2 d x u. 

. T - . 3 2 u du d 2 U rt 

4. If u = r n sin no. r 2 — -+r 1 — — = 0. 

dr 2 dr dO 2 

5. If u^ia-xy + ib-yy + ic-z) 2 ]-*, show that 

d 2 u d 2 u d 2 u 



if (1) u = x\ogy, 



dx 2 dy 2 dz' 



= 0. 



6. If u = f(y + ax)+F(y — ax), show that 
indicating any continuous functions. 



dx 2 



a' 



fi 2 u 
dy'' 



, / and F 



Successive Total Differentials. 
233. To find d 2 u. We have (§ 45) du=d x u+d y it, or 

7 du 7 du , 
au=^— dx-\-^— dy, 
dx dy * 

f du\ , du , rt , /du\ , du 



a) 



whence d 2 u=d l^—) dx-\-^- d 2 x-\-d (^— ) dy+^r- d 2 y. 

\dx/ dx \dy) u dy u 

Find d ( ^— ) and d (-7- ) by substituting ^— and ~- 

\dx/ \dyl J dx dy 

for u in (1). The final result is 



du 
dy 



d u d u d 2 u du du 

d 2 u=^—z dx 2 +2^ — ;=— ■ dx dy-\-^~— 5 dy 2 +^— d 2 x J r -^- d 2 y. 

dx z ox oy oy z ox oy 

d s u may be found in a similar manner. 



254 INFINITESIMAL CALCULUS. [Ch. XLIII. 

Ex. If u = c is a plane curve, du = and d 2 u = 0. If also d 2 x = 
(i.e., if x is the independent variable), show that 

d 2 u /du\ 2 d 2 u dudu d 2 u /du\ 2 

d 2 y dx 2 V yl dx dy dx dy dy 2 ^dx/ 
dx 2= ~ /^A 3 

\dy/ 



Extension of Taylor's Theorem. 

234. If in the formula of Taylor's Theorem, § 207, we write 
~ — , j. 2 , ... for f(x), f'{x), . . . , we obtain as an equiva- 
lent form 

f( x+ h)=Kx)+ d J^h+^-^+... (i) 

Let f(x, y) be a function of x and y, and let x become 
x+h, y for the present remaining unchanged. Then, from (1), 

/(»+*, y) = f(x, y)+ ^f h+ ^l »+. . . (2) 
If now y becomes y + k, (2) becomes 

Kx+h> y + k ^f (x ^^^ 

and each term may be expanded by Taylor's Theorem as 
follows, using u for f(x, y): 

df(x,y+k) h _ L ' dy j —^ U h-L ^ 2u hkj- 

lv ^7 lb ^Z IV I ^Z ^T lilv ~\ • . . • 

x da; ore do; or/ 



3 2 /(a-, y + k)h 2 d 2 [u + . . .] /i 2 3 2 w h 2 



dx 2 2! 3a; 2 2! 3a; 2 2! 



|T~ • . • 9 



234.] EXTENSION OF TAYLOR'S THEOREM. 255 

whence (3) becomes 

f(x + h, y + k) = f(x, y) + [_^ x h + ^ y k ~] 

If DE=—h + ^k, the form of (4) is the same as 

ox dy 

f(x + h,y + k) = u+Du+ — J r-^j +. . . 

D 2 Z) 3 






A similar result would apply to functions of three or more 
variables. 

Ex. 1. Euler's theorem on homogeneous functions. Def. A func- 
tion u or /(#, ?/) is said to be homogeneous and of the degree 
n when f(mx,my) = m n J(x, y), where m is any number. For 
example: 2x 3 + y 3 , x 2 — xy+y 2 , (x 2 +y 2 )/(x 2 — y 2 ), (x — y)/{x 2 +y 2 ), 
ax^ +by$. 

Letra = l+r. Then 

f(x+rx, y+ry)=(l+r) n f(x, y). 

Expanding the first member by Taylor's Theorem and the 
second by the Binomial Theorem, 

du dv\ ( Jd 2 u rt d 2 u d 2 u\ r 2 



l du dv\ ( jd'u _ d'u d'u\ r £ 

u+ [x— +y — ) r-Y lx 2 —-^+2xy Y — -) — + . 

\ dx u dvJ \ dx 2 u dxdu du 2 ) 2\ 



ey* it 



= [l+nr+n(n — 1)— + . . .]u* 



Equating like powers of r, 



du du 
x — Yy — = nu t 
dx J dy 

d 2 U d 2 u d ^ii 

x 2 —+2xy—- + y 2 —=n(n-l)u. 
dx 2 dx dy dy 2 



256 INFINITESIMAL CALCULUS. [Ch. XLIII. 

The results may evidently be extended to higher derivatives, 
and to functions of three or more variables. 
2. If u is homogeneous, show that 

d 2 u d 2 u . ^.du 

x — -+y = (n — 1)— , 

dx 2 dx dy dx 

d 2 u d 2 u , ^du 

x \-y — =(n — 1)— . 

dx dy u dy 2 v dy 

Maxima and Minima from Taylor's Theorem. 

2 35« By the aid of Taylor's Theorem we may verify and 
extend the conclusions- of Chapter XVII for maxima and 
minima. 

If a is a value of x for which any function f(x) is a 
max. or a min., and h any small quantity, it is plain that 
f(a + h)—f(a) and f(a — h)—f(a) must have the same sign, 
viz., + for a min. and — for a max. Now 

and }(a-h)-f(a)=-r(a)h+r(a)~-r(^ l + - ■ ■ 5 

and by taking h small enough the sign of the right-hand side 
will depend upon that of the first term which does not vanish. 
Hence there cannot be a max. or a min. unless /'(a) = 0, and 
there will then be a max. if /"(a) is — and a min. if /"(a) is + . 
But if f"(a) also = 0, there cannot be a max. or a min. unless 
f'"(a) also = 0> and there will be a max. or a min. according 
as p v \a) is — or +. It will thus be seen that there cannot 
be a max. or a min. unless the first derivative which does 
not vanish is of an even order, and that /(a) will be a max. 
or a min. according as this derivative is — or + . 

For a similar reason, from § 234 (4), a function u or 
fix, y) of two independent variables is a max. or a min. 



235.] 



MAXIMA AND MINIMA 



257 



for values a and b of the variables if a and b satisfy du/dx=0 
and du/dy = Q, and at the same time 



f U 2h 2 + 2 Vu hk A2 

Ox 2 dx dy dy 2 



(1) 



is not zero, and is in sign independent of the values of h and 
k. These conditions are satisfied if 



/ d 2 u \ 2 . 

\dx dy) 



d 2 ud 2 u I d 2 u 
dx 2 dy 2 \dxdy> 



is +. 



For, (l)^Ah 2 + 2Bhk + Ck 2 = 



(Ah + Bk) 2 +(AC-B 2 )k 2 
A 



and .'. has the same sign as A if AC — B 2 is +. 

TT .- du _ du _ , d 2 ud 2 u ( d 2 u \ 2 . 

Hence it ~ = u, ~- = 0, and ^-^ ^—15 — I ~ — ^- ) is + , u is 

Xn r \ox dy/ 



dx 



dy 



dx 2 dy 

d 2 u 
a max. or a min. according as ~— 5 is — or +. 

or 

Similarly for a function of three independent variables 
we must have du/dx=0, du/dy=0, du/dz=0, to solve for 
x, y, and z. 

Ex 1. u = x +xy +y 2j rx— 2y +4. 

du/d£ = 2a;+?/-hl, du/dy=x + 2y — 2. 

Putting these = and solving for £ and y we get x 
which make u a min., viz., If. 

2. The max. value of (2ax — x 2 )(2by — y 2 ) is a 2 b 2 . 

3. The max. value of (x — l)(y — l)(x +y — 1) is ^V« 

4. The min. value of x z +y 3 — 3axy is —a 3 . 

5. The max. or min. value of ax 2 +2hxy + by 2 +2gx-\-2Jy + c is 

a h g 



= 5 
31 



h b f 
9 f c 



a h 
h b 



6. Find a point such that the sum of the squares of its dis- 
tances from any number of given points (a,, 6 2 ), (a 2 , 6 2 ), . . . may 

/ 1 1 \ 

be a mm. Arts. ( — Ia y — lb) . the centre of mean position, 

\n n I 



258 



INFINITESIMAL CALCULUS. 



[Ch. XLIII. 



7. Given r x = a x x + b x y + c u r 2 = a 2 x + b 2 y + c 2 , . . . , show that the 
values of x and y which make r 1 2 +r 2 2 +r 3 2 + . . . a min. are ob- 
tained by solving the equations 

x I (a 2 ) +yl(ab) + l(ae) = 0, 
xl(ab) +yZ(b 2 ) + l(bc) = 0. 

These are the normal equations in the method of Least Squares. 

8. To make with the smallest possible amount of sheet metal 
an open rectangular box of given volume, show that the length 
and breadth must each be double of the depth. 

9. To cut circular sectors from the angles of a triangle so as 
to leave the greatest area with a given perimeter, show that the radii 
must be equal. 



CHAPTER XLiV. 
DIFFERENTIAL EQUATIONS* OF THE FIRST ORDER. 

236. A differential equation is an equation containing one 
or more derivatives. The derivatives are usually represented 
by the corresponding differentials. 

The order of a differential equation is the order of the 
highest derivative in the equation. The degree of the equa- 
tion is the degree of the highest derivative when the equation 
is free from fractions and radicals affecting the derivatives. 

d^y dy 
Ex. t4 + 2 - +y=0 is of the second order and first degree. 
dx 2 dx 

(dv\ 2 dy 
--) +2- - — H 2/ =0 is of the first order and second degree. 

Partial differential equations are those which contain 
partial derivatives; other differential equations are called 
ordinary. 

237. Ordinary differential equations frequently appear in 
the statement of problems in Geometry, Mechanics, Physics, 
etc., but for our present purpose they may be supposed to 
arise from the elimination of constants. 

-n ^y y dy y 

Ex. 1. y = mx, ~=m = —. .*. -f=—. 

(J/X X (J/X X 

This is a differential equation of the first order obtained by 
differentiating, and eliminating the constant m. It may be 

* For further information relating to differential equations see Mur- 
ray's Differential Equations (Lorgmans), from which some of the 
examples of this and the following chapter have been taken. 

259 



260 INFINITESIMAL CALCULUS. [Ch. XLIV. 

called the differential equation of all straight lines passing through 
the origin. 

7 dy d 2 y 

2. y = mx + b, — =ra, -^— = 0. 
* ' dx dx 2 

.*. d 2 y/dx 2 = is the differential equation of all straight lines. 

Two constants, m and b, have been eliminated. The equation is 

of the second order. 

o , dy rt d 2 y 1 dy 

3. y = ax 2 + b, -- = 2ax, -^ = 2a = — . -f. 

dx dx 2 x dx 

d 2 y 1 dv 

.'. -7-^ = =-, an equation of the second order. 

dx 2 x dx 

The elimination of n constants requires n+1 equations 
viz., the original equation and n derived equations. Hence 
the order of the resulting differential equation is equal to 
the number of constants eliminated. 

Eliminate a, b, c from the following equations: 

4. y = ae mx + be- mx . Arts. d 2 y/dx 2 = m 2 y. 

5. y = a sin mx + b cos mx. d 2 y/dx 2 = —m 2 y, 

6. y = ax 2 +bx + c. d 3 y/dx 3 *=0. 

8. 2/ 2 =(x-c) 3 . 8(dy/dxY = 27y. 

9. y = ax 2 +bx. x 2 -r±-2x~ + 2y = 0. 

238. An integral or solution of a differential equation is a 
relation between the variables which satisfies the equation. 



Ex. ?/ = A cos x, y = B sinx, y = As in x+B cosx, y = a sin (# + 6), 

d\ 
dx' 1 



d 2 y 
y = a cos (x +6), are all solutions of the equation x^+2/ = 0. 



The solution which contains a number of arbitrary constants 
equal to the order of the given equation is said to be a com- 
plete integral or general solution. Particular solutions are 
those which may be obtained from the general solution by 
assigning values to the constants, 



238-240.] 



DIFFERENTIAL EQUATIONS. 



261 



Separation of the Variables. 

239. In some cases no special method of solution is re- 
quired. An algebraical rearrangement of the terms will 
cause the equation to take the form 

h(x)dx+f 2 (y)dy=0, 

and each term may be integrated. 

Ex. 1. 2x 2 y dy=(l + x 2 )dx is the same as 

rt , /l+x 2 \ 1 dx , 

2 y d y=\—-r) dx= -2+ dx - 

Hence, integrating, y 2 = —x~ l +x+c, 
where c may have any assigned value (see § 96). 

1 7707 dy dx 

?<, y dx—x dy = dx+ x 2 dy, or = — — . 

* J y-\ x 2 +x 

Integrating, log (y — l) = log x — log (x 4-1) +log c,* 

.*. y — l=cx/(x + l). 



P>. (x 2 + y 2 — y)dx + x dy = is the same as 

x dy—y dx 



k-;) 



-Jdx-'r 



x- 



= 0, or dx 4- 



(i) 



d *- 



1 + 



© 



= 0. 



Hence x 4- tan -1 (y/x) = c, or ?/ = £ tan (c — x). 

4. x 2 y dy+m dx = Q. Arts. y 2 = 2m/x+c. 

5. (x — y 2 x)dx + (y — x 2 y)dy = Q. x 2 +y 2 = x 2 y 2 +c. 

6. x dy= (x 3 +y)dx. y = %x 3 +cx. 

7. (x 2 y+x)dy + (xy 2 — y)dx = 0. xy+\og(y/x) = c. 



8. y dy=(^x 2 +y 2 — x)dx. 



y 2 = 2cx + c 2 . 



240. The separation of the variables is sometimes assisted 
by a substitution. The following is an important case. 

* In order to simplify the final result the constant may be written 
in the form log c, or in any other form which permits of any arbitrary 
value. 



262 INFINITESIMAL CALCULUS. [Ch. XLIV. 

Homogeneous equations. If the given equation is of the 
form 

/i(z, y)dx + f 2 (x, y)dy=0, 

where the functions are homogeneous in x and y, and of the 
same degree, let y=vx. In the new equation in terms of 
v and x the variables will be separable. In some cases the 
substitution x=vy may be simpler. 

Ex. 1. xy 2 dy= (x 3 +y 3 )dx. liy = vx, 

v 2 {x dv +v dx) = (1 + v 3 )dx, or v 2 dv = dx/x. 
.' . %v 3 = log ex, or y 3 = Sx 3 log ex. 

2. (x 2 —2y 2 )dx+2xydy = 0. Arts. y 2 = —x 2 log ex. 

3. (x 2 + y 2 )dx = 2xy dy. y 2 = x 2 +cx. 

4. y 2 dx + x 2 dy = xy dy. x = y /log cy. 

5. (x+y)dy + (x — y)dx = 0. 

Arts. tsni- 1 (y/x)+log\ // x 2 -\-y 2 = c. 

6. Show that the homogeneous equation 

f x {x 9 y)dx+f 2 (x, y)dy = 0, 

or /,(1, v)dx+f 2 (l, v)dy = 

dx / 2 (1, v)dv 

becomes — +7-7^ — r^ — m — n = 0- 

x /x(l, t>)+v/ 2 (l, v) 

7. Show that an equation of the form 

f 1 (xy)y dx + f 2 (xy)x dy = 
can be integrated by the substitution y = v/x. 

241. An equation of the form 

(ax + by + c)dx+ (a'x + b'y + c')dy =0 (1) 

is not homogeneous, but may be reduced to a homogeneous 
equation by the method of the following example. 

Ex.1. (3x-y-5)dx + (x+y + l)dy = 0. 

Let x = X+h, y=Y + k. Then, substituting, 

{3X-Y+3h-k-5)dX + {X + Y + h + k + l)dY~0. 



241,242.] DIFFERENTIAL EQUATIONS. 263 

Take h and k so that Sh-k-5 = and h+k + l = 0. .'. fc=»l, 
&= — 2. The equation is now 

(3X - F)^x + (X + y )dr = o, 

which is homogeneous. Let Y = vX. Then 

1+v , dX 

3 +v 2 X 

1 i» 

whence, ——tan -1 — = + i log (3 +v 2 ) +log X = c 

V3 V3 

„ F V-fc V+2 

Bnt -Z-i=A-i=i- 

— tan- 1 — 1±L_ + ilo g [3(a;-l) 2 + (2/+2) 2 ] = c. 



V3 \/3(x~l) 

Hence to solve an equation of the form (1), drop the c 
and c', solve the resulting homogeneous equation, and in the 
result substitute x — h for x and y — k for j/, where A and k 
are the roots of the simultaneous equations ax + by + c=0, 
a'x+b'y + c' =0. 

The method would fail if a'x + b'y=k(ax + by), k being 
any constant; but the equation could then be solved by 
the substitution v =ax + by and the elimination of y or x. 

Ex. 2. (3x-22/-5)dx + (2x-32/-5)di/ = 0. 

^4ws. (x+y) 5 (x — y— 2)=c. 
3. (2x-y)dx-(4x-2y-l)dy = 0. 

Ans. 3(x-2y)+log (3y-6x+2) = c. 

Exact Differential Equations. 

242. The result of differentiating f(x,y)=c or u=c is 
(§ 45) 

M dx + N dy=0, 

where M=du/dx and N=du/dy. 

dM d 2 u . dN d 2 u* 

Hence t^— = ~ ~ ■ and 



dy dydx 'dx dxdy 

., . 3 2 ^ 3 2 u >. OQO , . dM dN 

r>Ut ~ — ^-^ = ^ — ^-— (§ ZoZ). . . -^— * — - — . 
oy ox ox oy oy ox 



264 INFINITESIMAL CALCULUS. [Ch. XLIV. 

Conversely, if M dx + N dy=0 is an equation such that 
dM/dy=dN/dx the equation is an exact differential equa- 
tion, i.e., one obtained directly by differentiating without 
further change. 

The re-integral of M dx contains all the terms of u except 
those which are independent of x. Hence to integrate an 
exact equation, integrate M dx with regard to x, integrate 
with regard to y those terms of N dy which contain y only, 
and put the sum of the results equal to a constant. 

Ex. 1. (4x~ + 6x 2 y)dx + (2x 3 -2y)dy = 0. 

Here dM/dy = 6x 2 , and dN/dx = 6x 2 . 

Hence the solution is x*+2x s y — y 2 = c. 

2. (2-2xy-y 2 )dx-(x+y) 2 dy = 0. 

Ans. 2x — x 2 y — xy 2 — \y z = c. 

3. (x 3 +y)dx+x dy = 0. ix 4 +xy = c. 

243. Integrating factor. After forming a differential equa- 
tion the result can sometimes be simplified by dividing by 
a variable factor. Conversely, a differential equation nay 
sometimes be made exact by multiplying by a factor. 

Ex. 1. (1 +xy)y dx + (1 — xy)x dy = is not exact, since dM/dy = 
\-\-2xy and dN/dx = l— 2x. Multiplying by l/x 2 y 2 the equation 
becomes 



(— +-)<fo+(— 5 )dy = 0. 

\xhi xl \xv l vi 



*y xi \xy* y 

Here dM/dy= —l/x 2 y 2 and dN/dx= —l/x 2 y 2 , hence the equa- 
tion is exact. The solution is therefore 

h log x — log y = c. 

xy 

1 y 2 1 

2. (x 2 +y 2 + l)dx — 2xy dy = 0, factor — . Ans. x =c. 

x l xx 

x + c 

1 1 T" 

3. 2(x dy+y dx) = xy dx, factor — . y =—e 

xy x 



243, 244.] 



DIFFERENTIAL EQUATIONS. 



265 



244. Linear equations of the first order. A differential 
equation is linear when it contains the first power only of 
the function and its derivatives. The linear equation of the 
first order is of the form 



dy 
dx 



+ Py=Q, or dy + Py dx=Q dx, 



where P and Q are independent of y. Of this equation e 
is an integrating factor. 

/Pdx fpdx fpdx 

. dy + y . e J . P dx=er Q dx 

(/Pdx) I Pdx 

ye J ) =e J Q dx. 

[Pdx I fPdx~ , 

.'. ye J = \ e Qdx + c. 

dy 
Hence the solution of -j-+Py=Q is 



/Pdx 



-/Pdx I [ fPdx n \ 

y=e J ( tV Q dx + cj . 



Ex. 1. -f- y = e x x n . 

ax x 



fpdx 

—n • p*J = rf—n 

• • • O tK/ • 



\P dx = — n log x = log x 

m 

.\ y = x n l e x dx + c) =x n (e x +c). 



dy ny a 
dx x x n 



Arts. y = 



ax+c 



x n 

3. dy/dx + y = e~ x . y = e~ x (x+c). 

4. x dy/dx = 2y+x + l. y= — x — %+cx 2 . 

5. dy/dx+y cos # = sin x cos x. y = sin x — 1 +ce~ sinx . 



6. *,*-,-* ov «£>-* 



1. 



p, c?v 1 — 2x 

7. / + — — 2/-1. 



Ans. i/ = Vex — x log #. 
2/ = x 2 (l +ce x ~ l ). 



266 INFINITESIMAL CALCULUS. [Ch. XL1\ . 

8. (l+x 2 )dy = (a+xy)dx. Ans. y = ax + cVl+x 2 . 

_ dy _ . 

9. -^-+ay = b smmx. 
dx 

Ans. ?/ = — -(a sin mx—m cos mx) +ce~ a:c . 

245. Bernoulli's equation. An equation of the form 

where P and Q are independent of y, may be made linear by 
the substitution —r=z. It is best to divide through by 

yn-l & J 

if 1 before substituting. 

^ ., dy , . 1 % 1 1 log x 

Ex. 1. x-f +j/ = t/ 2 log«, or - *+— .— =— — . 
ax y 2 ax x y x 







Let — 

y 


= z, then 

• 
• • 


1 

y 2 

dz 
dx 


dy = 

z 

X 


= dz 


log X 

X ' 










which is 


linear. 


Solving, 
























z = log X + 1 + 


ex. 


• 
* • 


11 — 


1 










log 2 


• + 1 + ex ' 




2 


(1- 


ax 


-#2/ = 3:r2/ 2 . 








Ans. 


y- 




1 






cVl 


— X 2 


-3 


3 


dx 


±xy=x l 


{ 2/ 3 . 










y- 




1 






Vl+x 2 


+ ce x2 



Equations of the First Order but not of the First Degree. 
246. Let dy/dx be called p. 
If pcssible solve the equation for p. 

Ex. 1. p 2 — (x — Sy)p — 3xy = 0, or {p — x)(p+3y) = 0. 
The equation is satisfied if 

p — x = 0, or p + 3y = ; 

i.e., if -r-x = 0, or - 2/ + 3/y = 0; 

9 dx dx ' * ' . 



245-248.] DIFFERENTIAL EQUATIONS. 267 

or, integrating, if 

y — %x 2 +c = 0, or y + ce~ 3x = 0* 

These equations may be regarded as the solutions of the given 
equation, or they may be combined into 



(y — %x 2 + c)(y+ce- 3x ) = 0. 



2. p 2 -9p + l8 = 0. 

3. p 3 = ax 4 . 



Ans. (y-6x + c)(y-3x+c) = 0. 
343(1/ + c) 3 = 27ax 7 . 



247. When it is not possible or convenient to solve for p 
we may be able to solve for y, then, differentiating through- 
out and substituting p dx for dy, obtain a new equation in 
p and x which we may be able to integrate and thus find 
the relation connecting p and x. From this result and the 
given equation we may be able to eliminate p and thus 
obtain the relation connecting x and y, or, if this elimination 
is not convenient or possible, x and y may be left in terms 
of p as a third variable. 

Ex. 1. p 2 x — 2py+x = 0, or 2y = px+x/p. 
Differentiating, substituting p dx for dy, and reducing, 

dp/p = dx/x, .'. p = cx. 
Hence, substituting in the given equation, 2y = cx 2 + — . 



2. p 2 —py + 1=0. 



1 



1 



Ans. x = —- '-f-log p+c, y = p + —. 



2p 



V 



248. Instead of solving for y we may be able to solve for 
x, then differentiate throughout and substitute dy/p for 
dx, and proceed as above. 



Ex. 1. p 2 -px + l- 

2. p 2 y + 2px = y. 

3. p 3 -p 2 x + 1=0. 



0. Ans. x=p+—, y = hp 2 — log p+c. 

r 

y 2 = 2cx+c 2 . 



1 

p 2 



p 2 2 

2 p 



268 INFINITESIMAL CALCULUS. [Ch. XLIV. 

249. Clairaut's equation. Singular solution. The method 
of § 247 is applicable to Clairaut's equation, 

y=px+f(p). ■ (1) 

Differentiating and substituting p dx for dy, 

dp[x + f'(p)]=0. 

From dp—0 we have p=c, and substituting in the given 
equation, 

y=cx+f(c), (2) 

the general solution. 

The equation is also satisfied if x + f f (p) =0, and eliminating 
p from this and the given equation we have another solution 
which is not contained in the general solution and which 
does not contain any arbitrary constant. Such a solution 
is called a singular solution. 

The general solution (2) represents, for various values of 
c, a family of straight lines. The singular solution represents 
the envelope of these straight lines. For the envelope of 
the family of lines is obtained (§ 157) by eliminating c from 

y =cx + f(c) and 0=x + f'(c), 

the same equations (with c instead of p) as those from which 
the singular solution is obtained. 

Ex. 1. y = px + a/p. The general solution is y = cx+a/c. Also 
x J \-f'(p) = isx — a/p 2 = 0. .*. p 2 = a/x. Substituting in the 
given equation we obtain y 2 = 4:ax, the singular solution. 

2. Find the singular solution of y = px-\-p 2 . Arts. x 2 +4y = 0. 

3. Find the general and singular solutions of 

y = px + a\^l +p 2 . 

Arts. y = cx + aVl +c 2 , . x 2 +y 2 = a 2 . 

4. Solve y= — xp+x 4 p 2 . Leto^z -1 . Arts. y = c/x+c 2 . 



r 



249.] DIFFERENTIAL EQUATIONS. 269 



Examples. 

1. dy/dx+y cot x = 2 cos x. Arts. y = sin x+c cosec x. 

2. x 2 dy + x 2 y 2 dx+4:dy = 0. y~ 1 = x — 2 tsur-^x + c. 

3. p 2 = px — y. y = cx — c 2 . 

4. px 2 + y 2 = xy. x = y(c +logx). 

5. (2x 2 +4:xy)dx + (2x 2 — y 2 )dy = 0. 2x 3 + 6x 2 y—y 3 = c. 

6. px + y = x 3 y Q . y- 5 = §x 3 +cx*. 

7. (x 2 — y 2 +2x)dx = 2ydy. x 2 — y 2 = ce~ x . 

8. (2ax+hy+f)dx + (hx+2by+g)dy = 0. 

ax 2 + 6i/ 2 + fon/ +fa+gy == c* 

9. 2xydx + (y 2 — 3x 2 )dy = 0. x 2 — y 2 = cy 3 . 

10. x dx dy = y dx 2 +2 dy 2 . cx = c 2 y + 2. 

11. x 2 p 2 = 2xyp+3y 2 . (xy — c)(y — cx 3 ) = 0. 

12. y 3 dx = (2x 2 + Sxy 2 )dy. 2xy+y 3 = cx. 

13. (y — a)dx=(x 2 +x)dy. (x + l)y = a + cx. 

14. x 2 {y — px) = p 2 y. Let y 2 = v, x 2 = z. y 2 = cx 2 -\-c 2 . 

15. Find the curve in which the subnormal is constant and = a. 
The condition is that ?/ dy/dx = a. 

Ans. The parabola ?/ 2 = 2aa; + c. 

16. Find the curve in which the subtangent is constant and = a. 

Arts. y = ce . 

17. Find the curve in which the perpendicular on the tangent 
from the foot of the ordinate is constant and = a. 

Ans. The catenary y = a cosh (x+c) /a. 

18. Find the curve in which the area bounded by the curve, 
two ordinates, and the #-axis is proportional to the length of the 
bounding arc. 

[y dx = dA = d(as) = a\ // dx 2 +dy 2 ]. 

Ans. The catenary y = a cosh (x+c) /a. 

19. Find the curve in which log s = x. 

Ans. y = \/e 2X — 1 — sec~ 1 e x + c. 

20. Find the curve in which <b = \d (§ 136). 

Ans. The cardioid r = c(l — cos 6). 

21. To find the amount, at compound interest due and added 
to the principal all the time, of a sum of money P, in t years, at 
rate r per unit per annum. 

Let x be the amount at time t. When the interest is due at the 



270 



INFINITESIMAL CALCULUS. 



[Ch. XLI V 



end of an interval At, xrAt = interest = Ax. Hence in the present 

case xr = £ Jt ^ (Ax/At) = dx/dt. 

Integrating, log x = ri + c. But x = P when 2 = 0. .*. logP = c. 
." . log x = rt+ log P, or x = Pe rt . 

For example, for 5 years at 4 per cent per annum, e r * = e' 04 x 5 = 
1*2214. The result may be compared with 1+^=1*2000, the 
amount at simple interest, and (1 +r) l = 1*2167, the amount at 
compound interest due yearly. 

22. Orthogonal trajectories. By giving different values to a 

in the equation x 2 -\-y 2 = 2ay we have 
a family of circles touching the x- 
axis at the origin. Let it be re- 
quired to find another family of 
curves all of which cut all of the 
given curves at right angles. Such 
curves are called orthogonal trajec- 
tories of the given curves. 

Differentiating x 2 + y 2 = 2ay and 
eliminating a we have 




Fig. 133. 



2xy + (y 2 -x 2 ) y = 0, 



the differential equation of the given curves. At a point Or, y) 
where one of the required curves intersects one of the given curves, 
dy/dx of the given curve ■•= —dx/dy of the new curve, since they 
intersect at right angles. Hence the differential equation of the 
required curves is 

2xy-{if-x 2 )-— =0, 
dy 

a homogeneous equation of which the solution is 

x 2 -\-y 2 = cx. 

The required curves are therefore circles touching the ?/-axis 
at the origin. 

23. Find the orthogonal trajectories of the family of parabolas 
y 2 = 4ax. Ans. The ellipses 2x 2 -\-y 2 = c 2 . 

24. Find the orthogonal trajectories of 

(1) The rectangular hyperbolas x 2 — y 2 = a 2 . Ans. xy^c 2 . 

(2) The straight lines y = mx. 

(3) The curves y = ax n . Ans. The conies x 2 + ny 2 = c 2 . 



249.] DIFFERENTIAL EQUATIONS. 271 

25. Show that the differential equation of the orthogonal tra- 
jectories of a family of curves represented by a polar equation is 
found by substituting —r 2 d0/dr for dr/dO in the differential equa- 
tion of the given curves. 

26. Find the orthogonal trajectories of r m cos mO = a m . 

Arts. r m sin md = c m . 

27. Find the curves which make an angle 45° with the parabolas 
r(l +cos 0) = 2a. Arts. The parabolas r(l +sin 6) =2c. 

28. If E = the extraneous electromotive force in a circuit having 
resistance R and inductance L, 

di 
L It+ Ri=E, (1) 

where i is the current at time t. The equation is linear, and if 
E is constant the solution is 

. E -ft 
K 



If 1 = when t = 0, c= -E/R. 

R 

t 



E E -?- 



■'■ i = R'R e L ' 

The second term soon becomes very small; the current is then 
practically constant and = E/R, as if there had been no induction. 

For an alternating E.M.F. let E = E m sin nt, where E m is the 
maximum value, and the time of a period is 2n/n. The solution 
of (1) is now 

E - — t 

t = 7T^ — 77 -,CK sin nt — Ln cos nt) +ce L 
R 2 + L 2 n 2 

E m , . ——t 



\/R 2 +L 2 n 2 



sin {nt — 6) + ce L 



where d = t&vr l (Ln/R). 

In a short time the exponential term becomes very small and 
the current is represented by a harmonic function of the same 
period as the E.M.F. 

is the lag of the current behind the voltage. The impedance 
(amplitude of voltage -v- amplitude of current) is ^R 2 +L 2 n 2 . 



CHAPTER XLV. 
DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 

250. Equations of the second order must contain d 2 y/dx 2 
and may contain dy/dx, x, and y. 

We shall first consider equations which do not contain y, 
and then equations which do not contain x. It will be seen 
that such equations may be reduced to equations of the first 
order. 

251. Equations which do not contain y directly. Let 
p=dy/dx. The equation will be reduced to one of the 
first order in p and x. 

_ ... d 2 y dy dp 

Ex.1. 7 f -y=a;, or - —p = x. 
ax 2 ax ax 

The latter is a linear equation of the first order. Hence, § 244, 

p = e x ( e~ x x dx + c) = — (1 +x)+ce x 

m 

.*. y = p dx= — f(l + x) 2 + ce x + c lt 

j 

dhi (dy\ 2 1 

2 - d^ a (dx)- An S .y = Cl -- log (ax+c). 

d 2 y 1 dy 
3 'W+xdx^°' y=clogx+c x . 

d 2 y 2dy c , 



dx 7 x dx 

d 2 y (dy\ 2 ^ 



x 



dv/ [dy\ 2 ^ 
5 - £"' ~ (dx) = A * y = Cl+ log sec (x+c) - 



272 



250-252.] DIFFERENTIAL EQUATIONS. 273 

The same method may be applied to any equation in 
which y appears only in two derivatives whose orders differ 
by unity. 

d 3 y d 2 v d 2 y 

Ex. —^-^ = 2. First let -~ = q. Ans. y = T %(x + c)3 + c x x+c 2 . 
dx 3 dx 2 dx* 

If dy/dx is absent as well as y we may integrate directly. 
^ d 2 y ■ dij x 4 x 5 

The s:me method will apply to any equation of the form 

d n y/dx n =f(x). 

Ex.1. d 3 y/dx 3 = sin x. Ans. y = cos x + ex 2 + c x x + c 2 . 

2. d n y/dx n =(x+n)e x . 

Ans. y = xe x + c x x n ~ l + c 2 x n ~ 2 + . . . + c n . 

252. Equations which do not contain x directly. Let 

dy ,, d 2 y dpdy dp mu ' . 

-r = v, then -7-5 =-— -7- =# —-. Ine resulting equation is of 
ax ax^ ay ax ay 

the first order in p and j/. 

t? 1 d2 y 2 * d P 2 

Ex. 1. -r-r= —a z y, or * p -—= —a 2 y. 
dx 2 ™ ^ dy * 

.'. \p 2 = —\a 2 y 2j r const., or say p 2 = a 2 (c 2 — y 2 ). 

dy ,-= — b ^v 

.\ p or -^— = a\/c—y\ or — - = a ax. 

^ dx v *" \/c 2 -?/ 2 

.*. sin~ 1 i//c = aa;4-c 1 , or y = c sin (ax + Cj). 

The result may also be written 

2/ = A sin ax + B cos ax, 

where A and B are arbitrary constants. 
2. d 2 y/dx 2 = a 2 y. 
The result may be in any of the following forms : 

y = C6 a:r -f Cje _a:r , i/ = c sinh ax + c x cosh ax, 

2/ = c sinh (ax + ^1), ?/ = c cosh (ax + Cj). 

2dy d 2 y 

* Or without using p, multiply by 2dy; then ■ " 2 ■ = —a 2 2ydy, 

(XJU 



( 



3? I = —a 2 y 2 -\- const., etc. 



274 



INFINITESIMAL CALCULUS. 



[Ch. XLV 



^S+(g^°- 



Ans. y 2 = cx J rc l . 



4. v 



d 2 y t /dy 
dx' 



+ 



m -• 



,2_^,2 



T/^ar + cz+C!. 



The Operator D. 

253. Let Dy represent dy/dx, the symbol D indicating the 
operation of taking the derivative of y. If a is a constant, 
d(ay)/dx =a dy/dx, hence D(ay) =a Dy. 

Let D . Dy, i.e., d 2 y/dx 2 , be represented by D 2 y. Also let 
dy/dx — ay or Dy — ay be expressed by (D — a)y. Then D — a 
indicates the operation of taking the derivative of a function 
and subtracting a times that function. Hence, a and b 
being constants, 

(P-b) . {D-a)y=D{D-a)y-b{D-a)y 

=D 2 y — a Dy — b Dy + aby. 

Let this be written 

[D 2 -(a + b)D + ab]y. 

In the same way it may be shown that 

(D-a) . (D-b)y=[D 2 -(a + b)D + ab]y, 

and hence the order of the operations indicated by D — a 
and D — b does not affect the result, and the successive opera- 
tions may be replaced by an operation which is indicated by 

D 2 -(a + b)D + ab. 

254. Suppose Dy =X, where X is a constant or a function 
of x, and let the inverse operation by which y is obtained 
from X be denoted by D~ l . Then y =D~ 1 X. But dy/dx =X. 



y 



X dx + c. 



D~*X = 



X dx + c. 



253-255.] DIFFERENTIAL EQUATIONS. 275 

Again, suppose (D — a)y=^X y and denote the operation 
inverse to D — a by (D — a)~ l . Then y = (D — a)~ l X. But 
(D — a)y =X is the same as dy/dx — ay =X, and the solution 
of this linear equation is (§ 244) 

y=e ax ( e~ ax Xdx-\-c\. 

j 

.'. (D-a)- l X=e ax ( e~ ax Xdx + c\. 

j ' 

If X=0, {D-a)~ l 0=ce ax . 

Ex. 1. D~ l = c, D-'O^D- 1 .D~ 1 = cx+c i . 

2. D- 1 a = ax+c, D~ 2 a = ^ax 2 + cx + c 1 . 

3. (D-a)- l b= -b/a + ce ax . 

4. {D-a)- l x= (x+—) +ce* x . 

a \ a/ 

.5. (Z> — a) -1 £ n , n a positive integer, 

1 / nx n ~ l n{n — l)x n ~ 2 n\\ 

= —lx n + + i- +. . .+— 1 -hce ax . 

a \ a a 2 a n I 

,_ v , . a sin mx-\-m cos mx 

6. (D— a)- 1 smmx= — \-ce ax . 

a 1 +m l 

,_ . — a cos mx-\-m sin mx 

7. (D — a) -1 cosmx = - = \-ce ax . 

a 1 +m 2 

gnx 

8. (D-a)- 1 e nx = h ce ax when n ^ a, 

n — a 

and =£e a z-l-ce a £, when n = a. 

Linear equation of the second order with constant 

coefficients. 

255. The equation may be written 

where A and B are constants and X is a constant or a func- 
tion of x. 



276 INFINITESIMAL CALCULUS. [Ch. XLV. 

First suppose X to be 0. The equation is now 

^ + A^+By=0, or (D 2 + AD + B)y =0. 

Put D 2 + AD + B into the form of factors (D-a)(D-b). 
(Since a and b would be the roots of D 2 + AD + B =0 if D 
were a symbol of quantity, they may be said to be the roots 
of the auxiliary equation z 2 + Az + B =0.) 
The equation is (D — a)(D — b)y=0. 

.-. y = (D-b)~ 1 (D-a)- 1 = (D-b)- 1 ce ax 

=e bx ( \e~ bx . ce ax dx + c^\ (1) 

/ce (a—b)x v 

=^*( — -T- + c 1 j,if Ma, 



= -— T^ + Cie^. 
a — 6 

Since c/(a — b) may equal any constant, it may be repre- 
sented by c. 

.-. y=ce ax + c 1 e bx y if Ma. (2) 

But if o =a, (1) becomes e a * f c dx + ci) . 

.'. y=e ax (cx + c 1 ). (3) 

If a pnd b are imaginary, let them be m + ni, m — ni, where 
£==V — 1. Then (2) becomes 

iy __^g(m+m)a; i q g(m—ni)x == ^mx(Q^nix _i_ q g—nix\ 

=e mx [c(cos nx + i sin nx) +Ci(cos nx — i sin nx)], § 200, 
=e m [(c + ci) cos ft£ + (ci — C\i) sin n^]. 

Since c + Ci and ci — C\i may equal any two constants, they 
may be represented by c and c\. 

/. y =e mx (c cos nx + ci sin n#). (4) 



. 



256 257] DIFFERENTIAL EQUATIONS. 277 

256. Hence to solve D 2 y + ADy + By=0 find the roots of 
z 2 + Az-\-B-=0. If the roots are unequal real numbers a 
and b the solution is 

y =ce axj rCie bx . 

If the roots are complex numbers m + ni, m — ni, 
y =e mx (c cos nx + Ci sin nx). 

If the roots are equal numbers a, a, 

y =e ax (cx + ci). 

Ex.1. -^ + 3-"-l(h/ = 0. Ans. y = ce 2x + c.e-^ . 
ax 2 ax 

2. g-8*-0. y^c+c^. 

ax 2 ax 

3. 2" 2 S +2/ = ' y=e x (cx+c 1 ). 

d 2 v dy 

4. —? 2 +2-^+10y = 0. y = e- x (ccos3x+c 1 sinSx). 

d 2/ u 

5. ~- 2 = a 2 y. y = ce aX + c x e~ aX or =c cosh ax-\-c x sinh ax. 

6. —-•-= —a 2 v. y = c cos ax + c, sin ax. 

7. (D 2 +4D+5)?/ = 0. y = e~ 2x (c cosx + Cj sinx). 

8. (D 2 +4D+4)2/ = 0. 2/ ^-^(cz+cj. 

9. (D 2 +4D+3)2/ = 0. ^/^cg-^+c^-^. 

10. CD 2 +4D+2) 2 / = 0. i/ = 6~ 2:c (c6 V2:c + c 1 6~ V2:c ). 

11. (6D 2 -5Z)-6)?/ = 0. ^cei^+c^-f*. 

257. The same method may be extended to higher orders 
of linear equations of the form 

d n y A d n ~ l y 

dx n dx n ~ l * v > 

the coefficients A . . . K being constants. For every distinct 
real root of the auxiliary equation, 

z n + Az n ~ 1 + . . , + K=0, 



278 INFINITESIMAL CALCULUS. [Ch. XLV 

there will be a term of the form ce ax in the solution. If a 
occur twice the corresponding term will be e ax (cx + c{) y and 
if it occur three times the corresponding term will be 

e ax (cx 2 + ciX + C2), 
and so on. 

Corresponding to a pair of imaginary roots m + ni, m — ni, 
there will be a term e mx (c cos nx + c\ sin nx), and if the same 
pair occur twice the corresponding term in the solution will be 

e mx [(cx + Ci) cos nx+ (C2X + C3) sin nx], 
and so on. 

Ex.1. CD-l)(D-3) 3 y = 0. Arts. y = ce x + (c 1 x 2 +c 2 x + C3)e 3x . 

2. (Z) 2 +4) 2 ?/ = 0. Arts. y = fax + c^ cos 2x + (csX + c 4 ) sin 2x. 

3. (D 2 -4) 2 2/ = 0. y=(c 1 x+c 2 )e 2x + (c 3 x + c 4 )er 2 *. 

4. D 4 y = a 4 y. y ■= c x e aX + c 2 e~ ax + cz cos ax+c 4 sin ax. 

5. CD 2 -4Z> + 13) 2 7/ = 0. 

Ans. 2/ = e 2x \{c x x + c 2 ) cos 3a; + (C3X + c 4 ) sin 3a;]. 

258. Returning to the linear equation, 

d 2 y. Ay R Y 

suppose now that X is not zero. The equation is the same 
as 

(D-a)(D-b)y=X=0+X, 

... y^{j)-h)-^{D-a)- l {)+{D-b)- l {p-a)- 1 X. 

The first term is the solution of the given equation when 
X is 0. This therefore forms a part of the required solution; 
it is called the complementary function, the remainder of the 
solution being called the particular integral. The com- 
plementary function will contain two arbitrary constants, 
and as the complete solution of an equation of the second 
order cannot contain more, we need not introduce constants 
in finding the particular integral. If they are introduced 
they will simply reproduce the complementary function. 



■I 



258,259.] DIFFERENTIAL EQUATIONS. 279 

Ex. 1. (D 2 -a 2 )y = e aX . The c. f. is ce ax + ce~ aX . The p. i. is 



(D-a)- 1 (D + a)- 1 e" x =(D-a)- i e 



— ip—ax 



e ax % e ax& x 



e 2aX 1 

= (D - a)-^-^ . = (D- a)~ l e^ x 

2a 2a 



1 

= — — ax 

2a 



1 

e-ax e ax(i x = Kze aX x. 



Hence the complete solution is 

y = ce aX + c y e~ ax + xe a x /2a. 

\{D + a)- l (D-a)- l e aX gives - e ax x--—e ax . The last term is 

2a 4cr 

included in the c. f.; hence the results are equivalent.] 

d 2 v dy 

2. ^.+2-^+y = e 2x . Ans. y = e~ x (cx + c 1 )+he 2x . 
ax ax 

3. (D 2 -l)y = 5x+2. y = ce x + c 1 e~ x -5x-2. 

4. {D-l) 2 y = x. y = e x {cx + c 1 )+x-\-2. 

5. (D-a) 2 y = e aX . y^e^icx + c^+^e^. 

6. (D 2 -4:D+3)y = x. y = ce x + c 1 e 3x +%x + i. 

7. (D 2 -4D + 3)y = xe x . y = ce x + c 1 e 3x ~ie x (x 2 +x). 

8. (D 2 +a 2 )y = e* x . 

The p. \.= (D + ai)~ l {D -aiy'e^, i^V^l, 

pax pax pax 



= (D+ai)~ l 



a(l-i) a(l-i) .a(l+i) 2a 2 ' 
.*. y = c cos ax + Ct sin ax+e aX /2a 2 . 

259. The last equation, (D 2 + a 2 )y = e ax , may also be solved 
as follows : 

Differentiating, D(D 2 + a 2 )y=ae ax , 
and multiplying the given equation by a and subtracting, 

(D-a)(D 2 + a 2 )y=0, 

a linear equation with the second member zero. The solution 
of this equation is 

2/=c cos ax + Ci sin ax + c 2 e ax . 



280 INFINITESIMAL CALCULUS. [Ch. XLV. 

The first two terms are the c. f. of the given equation; 
hence c 2 e ax is the p. i., where c 2 is to be determined so that 
c 2 e ax may satisfy the given equation. Substituting c 2 e ax 
for y in the given equation we find c 2 =l/2a 2 . 

Ex. 1. (D 2 -l)y = x 2 . 

Differentiating three times, D 3 (D 2 — l)y = 0. The c. f. is 
{D 2 -l)- l = ce x + c 1 e~ x . The p. i. is D~ 3 = c 2 x 2 + c*x + c A . Substi- 
tuting this for y in the given equation we find c 2 = — 1, c 3 = 0, c 4 = — 2. 

.\ y = ce x + c 1 e~ x — x 2 — 2. 

The p. i. might have been found more quickly by treating 
(D 2 — 1) _1 as if it were developable by the Binomial Theorem. 
Thus (D 2 - 1)-^'= - (1 -D 2 )~ 1 x 2 = - (1 +D 2 + ...)x 2 =- (x 2 + 2). 

d 2 v 

2. — 2 + a 2 y = b sin nx. Differentiate twice, eliminate the right- 

(J/ X 

hand member and show that 

b 

y = c cos ax + c x sin ax -f — ; sin nx. 

a 2 — n 2 

If a = n, substitute —bx/2n for b/(a 2 — n 2 ). 

3. (D 2 -2D+5)i/=l. Ans. y = i + e x (c cos 2x + c x sin 2a:). 

4. (D — l) 2 y = x 2 . y = e x {cx J rc l ) + z 2 +4a; + 6. 

5. (D 2 -2D+5)2/ = sin2a;. 

2/ = tV (4 cos 2# -f sin 2# ) + e x (c cos 2 x + c x sin 2x) . 

Change of Variable. 

260. Equations cf the second order, like those of the 
first order, are sometimes made integrable by a change of 
variable. 

Change of the dependent variable. 

_ d 2 v dy 

Ex. 1. x 2 - T Ji 9 +x/-y = x*. 
dx 2 dx 

Let y = vx, then dy = x dv + v dx, d 2 y = x d 2 v + 2dv dx. 

Substituting, the equation becomes 

d 2 v dv 
dx 2 dx 



260,261.] DIFFERENTIAL EQUATIONS. 281 

whence (§251) y = \x z + ex + c x x- x . 

d 2 y fdy\ 2 

2. y— -+ I ■- I =1. Lety 2 = v. Ans. y 2 = x 2 + cx+c t . 

ax 2 \dxl 

3. d 2 y/dx 2 = a 2 x — b 2 y. Let a 2 x — b 2 y = v. 

Ans. b 2 y = a 2 x + c cos bx + c x sin bx. 

261. Change of the independent variable. First sub- 
stitute 

dx d 2 y — dy d 2 x 



dx s 



(i) 



for d 2 y/dx 2 (§ 70). The subsequent change will depend 
upon the quantity which is to be the independent variable. 
This may be y, or a third variable z, x or y being an assigned 
function of z. It must be remembered that the second 
differential of the independent variable =0. 

Ex.l. ^+2^+^=0. (2) 

ax' ax x' 

Substituting (1) for d 2 y/dx 2 , (2) becomes 

dx d 2 y - dy d 2 x dy ahj 
X "dx* + 2x dx + 1? ~ °- (3) 

Let x = l/z and take z as independent variable. Then dx = 
— dz/z 2 , d 2 x = 2dz 2 /z 3 . Substituting in (3) we obtain 

d 2 y 

- + a 2 y = 0, whence y = c cos az + c x sin az. 



dz 



a . a 

. \ y = c cos — 4- c x sin — 

X X 



_ d 2 y 2x dy y 

2. -— + ——--+ 2 \2 i==Q - Letx = tan^. 
ax 2 l-hx 2 ax (l+x 2 ) 2 

Ans. y=(c + c 1 x)/\ // l+x 2 . 

d 2 y /dy\ ^ /dv\ 3 

3. -7-j — x ( • j + e^ ( — ) =0. Make 2/ the independent variable. 

d 2 x 
The equation becomes - T — + x = ey. whence 

dy 2 

x = c cos y + c x sin ?/ + \e y . 



282 INFINITESIMAL CALCULUS. [Ch. XLY. 

4. (l-x 2 )- 1 X-x^ = 2. Let z = sin 2. 
dx 2 ax 



Ans. ?/= (sin _1 ^) 2 +c sin-^ + Cj. 
d 2 y jdy\ 2 (dy x 



5 - {1 - y2) d +y {£) ~ x (-l) =0 - Let ^ sin *> * to be 



the 



independent variable. Ans. x = cy + c 1 \/l—y 2 . 

6. Show that the "homogeneous linear equation" of the second 
order 

x2 B +a 4l +hy = }{x) 

d 2 y dy 

becomes "r: + (a-l)-f +by = f{e z ) 

dz 2 dz 

(a linear equation with constant coefficients) by the substitution 

x = e z . 

d 2 y dy c 

7. x 2 - J {+2x---2y = 0. Ans. y = cx+-±. 

dx 2 dx x 2 

d 2 y dy 

8. x 2 -— — x -+?/ = log x. y= (cx + 1) log x + ^x+2. 

dx u dx 

9. x 2 ^+x^+4:y = x 2 . 

dx 2 dx * 

y = c cos (log x 2 ) + c x sin (log x 2 ) -f %x 2 . 

10. (2+^)4^ +3(2 +x) d ¥ + y = 0. Let2+z = e 3 . 

dx 2 dx 

Ans. y=(2+x)- 1 [clog (2+aO+cJ. 

Examples. 

1. Find the curve in which the radius of curvature R is equal 
to and in the same direction as the normal N. 

D L W J (l+p 2 )i . A7 ds ., ,.. 

dx 2 dy 

Ans. The catenary y = c cosh (x+cj/c 

2. Find the curve in which R= —N. 

Ans. The circle (x-\-c x ) 2 J r y 2 = c 2 . 

3. Find the curve in which R = e x , it being given that p = 
when £ = qo. Ans. y = c — sec~ 1 e x . 



261.] DIFFERENTIAL EQUATIONS. 283 

4. An elastic string (or spiral spring) is fixed at one end and 
hangs vertically. A weight is attached at the lower end and 
descends a distance a to a position of equilibrium 0. It is then 
pushed down a further distance b(<a) and released. To find x y 
the distance of the weight from 0, in terms of the time t from 
the instant of release. 

It is assumed that the weight of the string is negligible, and 
that tension is proportional to extension (Hooke's Law). Hence, 
since mass X acceleration = force, the differential equation of the 
motion of the weight is 

d 2 x a + x d 2 x q 

m -= m g-mg—, or ^—^x. (1) 

The solution is 

x = c cos v'g/a t + c ± sin ^g/a t. 

When 2 = 0, x = b and dx/dt (the velocity) =0. Hence c = b and 
c x = 0. 

. \ x = b cos ^g/a t, 

the required result. The values of x are repeated in magnitude 

and sign after an interval T if \ // g/a{t + T) = \/g/a t + 2x, i.e., if 

T = 27t\/a/g. The motion of which (1) is the differential equation 
is therefore oscillatory (a,simple harmonic motion) and the periodic 

time==27rVa/<7. Notice that the total extension of the string at 
time t is 

a + b cos ^g/a t. 

5. A fine chain of length 2a is placed over a smooth nail, the 
difference of the lengths on the two sides being 26. Find x, the 
length of the longer side, in terms of the time t from the instant 
of release. 

The whole chain is moved by the weight of the difference of 
the two sides; hence, m being the mass per unit length, the differ- 
ential equation of motion is 

£^2/£ d 2 (x ci} o 

^cim-— = mg[x-{2a-x)\ ) or — — — = Z-( x -a). 
av at 2 a 



Ans. x*=a-\-b cosh ^g/a L 



284 



INFINITESIMAL CALCULUS. 



[Ch. XLV. 



Show also that the chain will leave the nail in time 



*Ja/q cosh-Ka/fr) and with velocity Vg(a 2 -b 2 )/a, 




Fig. 134. 

6. Damped vibrations. The differential equation of simple 
harmonic motion is (as in Ex. 4) d 2 x/dt 2 = — a 2 x. If a frictional 
or other resistance proportional to the speed is applied, the equa- 
tion is (say) 

<Py \ dx 

— == — a 2 x— 2m— , 
dt 2 dt ' 

where m is a positive constant. If m < a the solution is 

x = e - m ( c cos nt + c x sin nt), or =Ae~ mt sin (nt + B), 

where n=v / a 2 — m 2 , and A and B are arbitrary constants. For 
A = l, B = 0, m = 'S y and a = 1*6, 

x = e~\ u sin 1-57*, (Fig. 134). 
If m is increased the waves become longer and flatter, and 
when ra = a the solution of the equation is 

x = e~ 1 ' 6t (cx + c 1 ), 

which for = 1^^ = 0, is the dotted curve of the figure, the curve 
being asymptotic to the 2-axis. 

7. Find the distance x which a body of mass *m falls from rest 
in time t, assuming that gravity is constant, and that the resist- 
ance of the atmosphere varies as the square of the velocity. 

Let the resistance be k when the velocity = 1. Then 

d 2 x , /dx\ 2 



Arts. x = — logcoshvl - 1. 
dt 2 \dt) k s \m 



\di, 



8. A uniform beam is fixed horizontally at one end, loaded 
with a weight w per unit length, and subjected at C, the centre 
of the free end, to a vertical force P and a horizontal tensile 
force Q. The origin being at C and the a;-axis horizontal, the 
equation of the elastic curve is 

EI^ 2 -Qy=-Px-hwx\ 



261.] 



DIFFERENTIAL EQUATIONS. 



285 



where E and / are constants depending upon the material and 
form of the beam. Write the equation in the form 

(D 2 — a 2 )y = —bx—fx 2 , 



where a = 



Q_ 
EV 



h=— f= w 
EV 7 ~2EV 



The c. i, = ce aX + c 1 e- aX . The p. i. can be found by the method 
of § 259, but more quickly by treating D 2 — a 2 as if it were a symbol 
of quantity. Thus 

-(2) 2 -a 2 )- 1 (to+/a: 2 ) = - 2 (l--^)~ 1 (6^+/a; 2 ) 

a 2 \ a 2 / 

bx fx 2 2/ 
.'. y = ce ax + c x e- a *+-- +'— ■+-. 

a 2 a 2 a 4 

To determine c and c u x = when y = 0, 

V- = c + c 1 +2//a 4 . 

Also, since the tangent at the fixed end is horizontal, dy/dx = 
when x = the length I, very nearly. 

.'. = a(ce al -c x e-<*) +b/a 2 +2fl/a 2 . 

These two equations may be solved for c and Cj. 

9. If Q (Ex. 8) acts in the opposite direction show that 

bx fx 2 2 / 
y = c cos ax + c x sin ax + — + — — — . 



a" a 



10. Curve of pursuit. To find the path of a dog which runs to 
overtake his master, both moving with uniform speed, and the 
latter in a straight line. 

Take this line for ?/-axis. When the dog is 
at (x, y) on the curve, the distance of the man 
from the origin is y — px (the intercept of the 
tangent on the y-axis), and by supposition, 

dt dt y 

where k is a constant (speed of man ~ speed of dog). 




/. —xdp = k\ // l+p 2 dx 



286 INFINITESIMAL CALCULUS. [Ch. XLV. 

is the differential equation of the curve. The solution is 

cx 1+k c- 1 x 1 ~ k 

and =c 1 -f^c£ 2 — c _1 log x, ii k = l. 

The constants c and c x can be determined if the initial condi- 
tions are assigned. 

If the man starts from 0, the dog from any point A on the 
#-axis (see figure), and they meet at B, show that the length of 
the curve = AC + CB, where C is the middle point of OA. 



APPENDIX. 

» 

Note A. Partial Fractions. 

The algebraical sum * of certain fractions being given, it 
is required to find the fractions. 

Case i. When the factors of the denominator are all of 
the first degree and unequal. 

_ £ 2 +3x + l 

Ex. 



x(x-l)(x + 2)' 



The denominator indicates that this result may be obtained by 
the addition of three fractions whose denominators are x, x — 1, 
x + 2, respectively; our object then is to find the numerators. 
Call them A, B, CVf 

x 2 +3x + l ABC 

+rr^- (1) 



x(x-l)(x + 2) x x — 1 x+2 
Clearing of fractions, 

x 2 + Sx + 1 = A(x -\){x+2) +Bx(x + 2) +Cx(x - 1). (2) 

It is to be noticed that (1), and .'. (2), is to be an identity 
and therefore true for all values of x. Now, if we give any three 
successive arbitrary values to x in (2), we shall obtain three 
equations by solving which, as simultaneous equations, the quan- 
tities A, B, C may be found. The arbitrary values should, how- 
ever, be such as to render these three equations as easy of solution 

* The sum is assumed to be a prop r fraction (the numerator of 
lower dimensions than the denominator). If not, the fraction should 
be reduced to a mixed quantity. 

t The A, B,C are assumed to be independent of x. If one of them 
contained x the sum would not be a proper fraction. 

287 



288 APPENDIX. 

as possible. A little inspection will show that this will be accom- 
plished by giving x successive values which make x, re — 1, and 
x + 2 equal to 0, i.e., by making a; = 0, 1, and —2. We thus get 
from (2), 

when x = 0, l=A(-2), .\A=-%; 



5 . 

3 > 



when x=l, 5 = 5(3), .*. B 

when rr=-2, -1 = C(6), .\ C= ~i 



" re(rc-l)(rr+2) a: re — 1 x + 2 

1 5 



2z 3(x-l) 6(rr + 2)" 

Case 2 . When the factors of the denominator are of the 
first degree, but two or more of them are equal. 

l+3x 



Ex. 1. 



x{x + l) 2 ' 



The denominator shows that the partial fractions have denomi- 
nators x and (rc + 1) 2 and (probably) rc + 1. We therefore assume 

1+3Z A B C 

= + 



x(x+iy x x+i {x+iy 

.'. l+3x = A(x + l) 2 +Bx(x + l)+Cx. 

If z = 0, 1 = A, .'. A = l. 

If a? — — 1, -2=-C, .*. C=2, 
and B may be found by giving any value other than and —1 
to x y e.g., if x = l we have ("." A = 1, and (7 = 2), 

4=lX2 2 + #X2+2Xl, .*. £=-1, 

l+3rr 1 1 2 

7 + 



' x(x + l) 2 a; x + 1 (rr + 1) 2 * 
x + 2 ABC D 

2. TT-; — = 7 + 



(x+i)(x-iy x+i x-\ (x-iy (x-iy 

.'. x+2 = A(x-iy+B(x + l)(x-l) 2 + C(x + l)(x-l)+D(x + l). 

If re 1, 1=A(-2)V .'. A=-%. 

Iix = l,3 = D(2), .-. D-|. 



PARTIAL FRACTIONS. 289 

To gel B and C give x any two arbitrary values other than 1 
and —1; thus (remembering that A and D are found) if x = 0, 

2=\+B-C+% or B-C = h 

and if x = 3, 2B + C = ; hence from these two equations 

x+2 1113 



(x + l)(x-l) 3 8(3 + 1) 8(o;-l) 4(z-l) 2 2(z-l) 3 ' 

Case 3. When the denominator contains a quadratic 
factor which cannot be conveniently factorized. We now 
assume the numerator of the fraction with a quadratic de- 
nominator to be of the form Ax+B. This is equivalent to 
assuming tw r o fractions with denominators of the first de- 
gree and constant numerators. 

_ 1+x Ax+B C . . _ _ „,_ , x 

Ex. — ■ -=- -+-. .*. l+x = Ax 2 +Bx + C(l+x 2 ). 

x(l+x 2 ) 1+X 2 X 

If 3 = 0, 1 = C, /.. C-l. 

Ifz = l, 2 = A+B+2, .'. A+B = 0, 1 .-. A=-l, 

If s--l, 0=A-B+2, .-. A-£=-2, J 5 = L 

1+x -z + 1 1 

^ _ J — m 

x(l +X 2 ) 1 +X 2 X ' 

If the given denominator had contained the square of 1 +x 2 , 

we should have assumed an additional term — — . 

(1 +x 2 ) 2 

Be ides the methods explained in the above examples others 
may sometimes be employed with advantage. For instance, 
in the last example 

l + x=Ax 2 + Bx+C(l + x 2 ). 

Since the left- and right-hand sides are to be identical, the 
coefficients of like powers of x on the two sides must be equal ; 
we .*. have 1 =C, 1=B, 0=A + C, which give the same 
results as before. 



290 APPENDIX. 

Note B. Curve Tracing. 

1. In order to trace a curve accurately from its equation 
we must be able to express one of the coordinates in terms 
of the other, or both in terms of a third variable. When 
the rectangular equation contains terms of two degrees only, 
we may substitute mx for y and solve for x, and in this way 
obtain both x and y in terms of m. See foot-note, p. 53. 

2. The following suggestions and remarks may be found 
useful in curve tracing, in order to shorten or verify the 
work. 

(I) Examine the equation for symmetry. When the equa- 
tion remains unchanged if — y is substituted for y the curve 
is symmetrical with reference to the line y = (the x-axis), 
for if the coordinates (a, b) satisfy the equation, (a, —b) 
will also satisfy it. This will always be the case if the equa- 
tion contains only even powers of y. Similarly the curve 
is symmetrical with reference to the line x=0 (the y-axis) 
if its equation is not altered when x is changed into —x. 

If the equation is unaltered by changing x into — x and y 
into — y at the same time, every line drawn through the 
origin and terminated by the curve is bisected by the origin; 
for if (a, b) satisfy the equation, ( — a, —b) also satisfy it 
and the origin is the middle point of the line joining these 
points. The origin is then called a centre; e.g., in the curves 
y=x 3 , y=s'mx, etc. 

The curve is symmetrical with reference to the line y=x 
if the equation is unaltered when x is changed into y and y 
into x, e.g., x s + y s =3axy (Fig. 28) ; and it is symmetrical with 
reference to the line y = — x if we can change y into — x and 
xinto —y without altering the equation, e.g., inx s — y 3 =3axy. 

If in polar equations the substitution of — 6 for 6 does not 
alter the equation, the curve is symmetrical with reference 
to the initial line (e.g., in Figs. 88, 89, 90, 92); and if we 
may at the same time change r into — r and d into — with- 
out altering the equation, the curve is symmetrical with 



r 



CURVE TRACING. 291 

reference to a line through the origin perpendicular to the 
initial line (e.g., in Figs. 84, 85, 98). The origin is a centre 
when we can change r into — r without altering the equation 
(e.g., in Figs. 86,98). 

(II) Find the tangents at the origin (if the origin lie on 
the curve) and the shape of the curve near the origin (§§3 
and 4 below); also, if possible, the points of intersection of 
the curve and the ax^s, and the directions of the tangents 
at these- points; the points where the coordinates are maxima 
or minima; the points of inflexion; the asymptotes rectilinear 
or curvilinear, etc. 

(III) No straight line can meet a curve of the nth degree 
in more than n points, and therefore no tangent in more 
than n — 2 points besides the point of contact, no asymptote 
in more than n — 2 points at a finite distance and no line 
parallel to an asymptote in more than n— 1 points at a 
finite distance, no line through a double point in more than 
n — 2 other points, etc. 

3. The work of tracing a curve from its equation is often 
considerably lightened by obtaining a preliminary idea of the 
shape of the curve at certain points. 

When the origin is a point on a curve we can find the 
shape of the curve very near that point by expanding y 
into a series of ascending powers of x. Thus in Fig. 32, 
y = ±x(l + 2x)^ y and taking first the + sign we have by 
the Binomial Theorem, 

y =x(l + x— . . . ), or y=x + x< 



• • • 



The term x shows that when x is very small (and .'. the 
third and higher powers of x may be neglected) the curve lies 
above its tangent y =x both when x is + and when x is — ; 
in fact the curve is, for points near the origin on the branch 
touching y=x, shaped nearly like the parabola y=x + x 2 . 
Similarly on the other branch y = — x— x 2 +. . . ; hence this 
branch lies below the tangent on both sides of the origin. 



292 APPENDIX. 

Similarly we may show that in Fig. 34 the curve near the 
origin is shaped nearly like the parabolas y=x 2 , y = —x 2 . 

4. When it is not convenient or possible to express one 
coordinate in terms of the other we may proceed as in the 
following examples: 

Ex. 1. In the curve a 2 (y — x)(y + x)= — (y 2 +x 2 ) 2 , Fig. 27, con- 
sidering first the branch which touches y—x = ® (§52) we divide 
by a 2 (y+x) and write the equation in the form 

_(y 2 +x 2 ) 2 
V ~ X a 2 (y+xY (1) 

For points near the origin on the branch in question y is very 
nearly equal to x, and the fraction in (1) must be very small; 
we shall get an approximation to its value by substituting x for y; 
this gives 

y = x—2x 3 /a 2 , 

which shows that the curve lies below the tangent when a: isl- 
and above it when x is — . For the other branch we write the 
equation in the form 

(y 2 +x 2 ) 2 

y=-x- \ \ , (2) 

a 2 (y. + x) 

and remembering that y is nearly equal to —x we substitute — x 
for y in the fraction and get 

y= —x-\-2x 3 /a 2 , 

showing that the curve lies above the tangent when x is + and 
below when x is — . 

2. In the curve 3axy = x 3 +y 3 , Fig. 28, the tangents at the 
origin are y = Q and x = 0. Writing the equation in the form 

x 3 + y* 
Sax 

we observe that on the branch which touches y = (the rr-axis) 
y is nearly near the origin, and substituting this for y in the 







CURVE TRACING. 293 

fraction gives y=-x 2 /3a for the approximate form of the curve. 
For the other branch 

x 3 +y 3 



x 



Say 



and writing for x in the second member we get x = y 2 '/3a for 
the required approximation. Thus the curve is shaped near the 
origin like a pair of parabolas. 

3. Find the approximations to the three branches of the curve 
ay(y-V3x)(y + V3x) = x 4 , Fig. 36, near the origin. 

Arts, 6a(y — \ /r 3x) = x 2 , 6a(y + V3x) = x 2 y 3ay= —x 2 . 

4. Also of ay 2 (y-x)(y+x) = x 5 , Fig. 37. 

Ans. ay 2 =—x 3 , 2a(y — x) = x 2 , 2a(y+x)= — x 2 . 

5. Show from these approximations that the radii of curvature 
at the origin are \a and ±24a in Fig. 36, and 0, ±2^/2a in Fig. 37. 
(Cf . § 88, Ex. 2.) 

5. The asymptotes of a curve may be obtained by expand- 
ing y into a series of descending powers of x (see § 57) . When 
it is impossible or difficult to express one of the coordinates 
in terms of the other we may proceed in a manner similar to 
that of § 4 above, beginning, however, with the terms of the 
highest degree instead of those of the lowest. (See § 59.) 

Ex. 1. x 3 +y 3 = 3axy, Fig. 28. Here x + y is a factor of the terms 
of the highest degree, and we may write the equation in the form 

3axy .. . 

y=-x + - -— . (1) 

x L — xy +y* 

Now the infinite branch is in the direction of the line y— — x, 
and therefore when x is very large, y is nearly equal to — x; hence 
we shall get an approximation to the fraction in (1) by substituting 

— x for y; this gives 

y = — x — a> 

which is the nearest linear approximation to the 'curve, and is 
therefore the equation of the asymptote. Writing — x — a for y 
in the fraction will give a second approximation, viz., 

a 3 



294 APPENDIX. 

from which it appears that the curve lies above the asymptote 
whether a; is + or — . 
2. Find by this method the asymptotes of the following curves: 

(1) x 3 (y — x) = a{y z +x*). Arts. y = x+2a. 

(2) xy 2 (y — x) = y 3 — 2x 2 y + x 2 . x = l, y=l, y = x — l. 

(3) (x + 2y)(x-y) 2 = 6a 2 (x + y). x+2y=Q, x-y= ±2a. 

Examples. 

1. Trace the following curves: * 

(1) y = x(x 2 -l), (fe) y 2 = x 2 (x-l) y (11) x 5 + y* = a 3 , 

(2) y(x 2 -l) = x, (7) x 3 -y 3 = 3axy, (12) x{y-x) = ay 2 i 

(3) y(l + x 2 ) = x, (8) x* + y 4 = a 2 xy, (13) x(y-x) 2 = y\ 

(4) y 2 -x 3 (x + l), (9) x 5 +y f > = 2a 3 xy, (14) a 2 ?/(a;+2/) = z 4 . 

(5) 2/ 3t =x 3 (z-l), (10) a: 5 +2/ 5 = ax 4 , 

2. Trace the following polar curves: 

(1) r = a sin 20, (6) r = atan0, (11) r(0 2 -l) = a0, 

(2) rsin20 = a, (7) r 2 = a 2 d, (12) r0 2 = a(0 2 -l), 

(3) r = a sin 30, (8) r = a0 2 , (13) r(0 2 + l) = a0 2 , 

(4) rsin30 = a, (9) rd 2 = a, (14) r0 = tan 0. 

(5) r 2 = a 2 sin 30, (10) r(l + 0) = a0, 

Note C. Hyperbolic Functions. 

(For definitions and graphs of these functions see Ch. 
VIII.) 

1. The relations connecting the hyperbolic functions are 
similar to those connecting circular (trigonometrical) func- 
tions, and are easily proved by ordinary algebra, etc. Some 
of them are as follows : 

cosh 0=1, sinh0 = 0, tanh0 = 0, etc. 

cosh ( — x) =cosh:r, sinh (— x) =— sinhz, tanh (—x) = — tanhrr, 
etc. 

* Some of these examples are taken fiom Frost's Curve Tracing, 
to which the student is referred for further information on this sub- 
ject. 



HYPERBOLIC FUNCTIONS. 



295 



cosh 2 x — sinh 2 a; = 1 , sech 2 x = 1 — tanh 2 x, cosech 2 x = coth 2 x — 1 . 

cosh (x±y) = cosh x cosh 2/±sinh x sinh y, 

sinh (x±y) =sinh x cosh y±cosh x sinh y, 

cosh x + cosh y = 2 cosh \{x+y) cosh \{x — y), 

cosh x — cosh y =2 sinh ?(x + y) sinh §(^ — 2/)? 

sinh x + sinh y =2 sinh J (^ + 2/) cos h i( x ~~y)y 

sinh x — sinh ?/=2 cosh i(x + y) sinh §(# — y), 

cosh 2x=cosh 2 £ + sinh 2 :E, 

= 2 cosh 2 x —1 = 1 + 2 sinh 2 x, 
sinh 2x=2 sinh a; cosh x, 

tanh £±tanh y 



tanh (x±y) 



tanh 2x = 



l±tanh x tanh ?/' 
2 tanh x 



1 + tanh 2 #* 



2. The differentials, integrals, etc., are as follows: 



d sinh x = cosh x dx, 



d cosh x =sinh x dx, 



d tanh x =sech 2 x dx, 



cosh x dx =sinh x. 



sinh x cte=cosha;. 



sech 2 x do; = tanh x. 



d coth x = — cosech 2 x dx, cosech 2 x dx = — coth x. 

rf sech x = — sech x tanh # dx, sech x tanh xdx^ - sech x. 
dcosech.r= —cosechxcothxdx, cosechxcothxdx= — cosechx. 



cfsinn l - 



- , f ^ = S inh-i* = log ^ +Va:2+a ' 2 ' 

a Vx 2 + a 2 JVx 2 + a 2 a °\ a 



296 



APPENDIX. 



a cosh" 1 — =■ 



a Vx 2 -a 27 



- , = cosh 1 -=lo2; 

\/x 2 -a 2 a °\ a , 



1 1 i i **^ a ax I ,, i 
atanh -= — „, x < \a, 



a a 2 — x 2 



dx , , . N 1 , - a; 1 , fa + x 
,*—, -s(^|< 1«) = — tanh i-=_log i 



a^ — X' 



a 



a 2a 



a — x 



a cotn - = — oj # > a, 



a a 2 — x 2; 



x 



a 
adx 



ax ^ i I v i ,i _i 3/ i i /x -r a\ 

; -(x >a) = — coth 1 -=-—lo2; ( ) 

5 -x 2V ' ] y a a 2a 5 \x-a/ 



d sech * — = — 

a xV a 2 — x 2 



C dx. 1 , - x 1 , / x \ 

— - = seen i-=-log ( ) 

JxVa 2 -x 2 a a a Va+vV-x 2 / 



_ . - x a ax 

a cosech -1 — = — 



a xVa 2 + x 2 ' 

C dx 1 , , x 1 , / x v 

— = — cosecn" 1 - = - log ( ) 

JxVa 2 + x 2 a a a Va+va+x/ 



/v»0 /Y»5 

sinh x = x + ^-: + ^-.+ . . . 
o! 5! 

cosh x=l + —. + — + . . . 
Z\ 4 ! 

tanh -1 x^ x + tt + ^ +. . . 
6 5 

. _ _J_x 3 1 . 3x 5 1 . 3.5 x 7 

smh x-x 2 3 + 2.4 5 2 4.67 +### 



If isV-1, cos ix = cosh x, sin ix=n sinh x, cosh ix = cos x 
sinh ix =i sin x. 



— — 



HYPERBOLIC FUNCTIONS. 



297 



x 2 y 



3. At any point of an ellipse — + ^-=1 (Fig. 136) we may 



a< 



put x =a cos u } y=bsmu } since cos 2 w + sin u=l. In this 




Fig. 136. 



case u=2 area AOP/ab (see Ex. 14, p. 162); it also = the 
eccentric angle AOQ. 



y 



At any point of a hyperbola -g— ^=1 (Fig. 137) we may 
put x = a cosh u, y=b sinh u, since cosh 2 ^ — sinh 2 ^ = 1 . In 
this case u=2 area AOP/ab =log ( — h^), (see Ex. 14, p. 140). 





fay)' 



Fig. 137. 



Fig. 138. 



If 6 = a the ellipse becomes the circle x 2 + y 2 = d 2 , and the 
hyperbola the equilateral hyperbola x 2 — y 2 =a 2 . Also u is 
in both cases the measure of the area of the sector AOP 
when \a 2 is taken as unit area. The circular and hyperbolic 



298 APPENDIX. 

functions may be defined in terms of u, and correspond- 
ing to 

y/a=sinu, x/a=cos u, y/x=t&riu, etc., - 

for the circle ; we have 

y/a =sinh u, x/a =cosh u, y/#=tanh u, etc., 

for the equilateral hyperbola. 

4, Gudermannian. If 2= log tan (\n + %0) or log (sec + 
tan 6), is called the gudermannian of z (gd z) and z =gd~ 1 0. 

Since e*=sec # + tan 0, .*. e~ z =sec — tan 0. Hence 
cosh 2=sec 6, sinh 2=tan 6, tanh 2=sin 0, etc. Thus if is 
tabulated for values of z the hyperbolic functions may be 
obtained from a table of circular functions. 

Differentiating one of the relations connecting and z, 
we obtain dd =sech z dz, or dz =sec dd. 

.*. d(gd z) =sech z dz, and d(gd -1 #) =sec dd. 

The inverse gudermannian is also written X{0) and called 
the lambda function, i.e., 

X{6) =log tan {\n+\0) =log (sec # + tan 0). 
Ex. Show that tanh ^ = tan \ 6. 

5. In the equilateral hyperbola x 2 — y 2 =a 2 , Fig. 138, let 
u be, as in § 3 above, the area of the sector AOP in terms 
of |a 2 as unit area. From the foot M of the ordinate MP 
draw MB tangent to the circle x 2 + y 2 =a 2 . Then x/a =cosh u 
and also=sec 0. .-. 5 is the gudermannian of u. It may 
also be proved that (1) MB=y, (2) tanh w=tan AOP =sin 0, 
(3) the line through parallel to BP bisects both sectors 
AOB, AOP, and the chord AP. 



MECHANICAL INTEGRATION. 299 



Note D. Mechanical Integration. 

1. Sign of an area. Let a straight line AB of constant 
length move in a plane to any other position A r B r , thus 
describing or sweeping out an area. Let it be agreed that 
any portion of AB describes a positive or a negative area 
according as, when viewed from A, it moves toward the 
left or the right. Thus the whole area is + in Fig. 139, 
- in Fig. 140, while in Fig. 141, BOB' is + and AOA' is -. 

B' 



b a 





A' 

Fig. 140. Fig. 141. 

2. Measurement of the area. AB can be moved to any 
other position A'B f (Fig. 142) by (1) a translation to A'D, 
during which the points in AB describe straight lines, and 
(2) a rotation about A', during which the points describe 
arcs of circles. The middle point M of AB moves first 
to F and then to M'. Take ME perpendicular to AB. 
The area of the parallelogram AD =AB . ME, the area of 
the sector DA'B' =A'D . FM r ) hence the whole area = 
AB(ME + FM'), i.e., ABXthe total normal displacement of 
its middle point. 

Suppose a wheel to be attached to AB at M with its 
axis in the direction AB, and that suitable graduations 
record the number of revolutions and parts of a revolution 
which the wheel makes. Let n be this number, i.e., the 
change cf reading of the recording circles between the time 
of starting and any subsequent time. Let c = the length of 
the circumference of the wheel; then en is the distance 



300 



APPENDIX. 



rolled through by a point in the circumference of the wheel. 
Take b for the length of AB. During the motion of transla- 
tion the wheel rolls over ME and slides through EF, during 
the rotation it rolls over FM'. Hence the total normal 
displacement of M =cn, and the total area described by 
AB=bcn. 

If, as in Figs. 139, 140, 141, A and B describe curves, 
imagine the motion to be a combined translation and rota- 
tion with infinitesimal displacements, any of which may 
be negative as well as positive. Then the rate at which the 
area is described is bXrate of normal displacement of M, 





Fig. 143. 



and hence the total resultant area =6 X total normal displace- 
ment of M =bcn. 

3. Consider now the effect of putting the wheel at any 
point L in AB, Fig. 142. The distance rolled over by the 
wheel is now LG+HL', and hence the normal displace- 
ment of M =cn+FM'-HU=cn+(A'F-A'H)d=cn+hd, if 
LM=h. In a circle (Fig. 143) of radius h draw OP, OF' 
parallel to AB, A'B'. Then hd=PP'. Hence the area 
described by AB=b(cn + PP'), and if AB moves to any 
new position to which OQ is parallel, the resultant area swept 
out =b (en + PQ). If AB moves about, turns back, and 
finally returns to its first position, Q returns to P and the 
resultant area =bcn, as if the wheel were at M. But if AB 
makes a complete revolution and returns to its first position 
the resultant area =6(cn+2^/i). 




MECHANICAL INTEGRATION. 301 

4. Closed curves. Let a straight line move so that its 
extremities describe any closed curves. Then in all cases 
the area swept out by the line will be equal to the arith- 
metical difference of the areas of the curves described by 
its extremities. 

When the areas are without one another, one will be 
described on the whole positively and the other on the whole 
negatively, while the area be- 
tween them, if swept out at all, 
will be swept both positively and 
negatively. When they inter- 
sect, the common portion, in so far 

as it is swept at all, will be swept 

. Fig 144 

both positively and negatively; 

the rest as before. When one curve lies entirely inside the 

other, the portion of the foimer which is swept at all will 

be swept both positively and negatively. 

5. Amsler's polar planimeter consists essentially of two 
bars, CA, AB, hinged at A, a recording wheel being attached 

to AB at any point L. C is fixed 
while B is moved round a curve. 
But if A is constrained to move 
along any line — whether straight or 
curved — without enclosing any area, 
Fig. 145. the area of the curve traced out by 

B is equal to the resultant area 
swept out by AB, and hence will be ben if AB returns to 
its starting place without making a complete revolution. 
But if C lies inside the curve described by B, AB makes 
a complete revolution, and the area of the curve described 

by B 

=b (en +2nh) + circle described by A 

tl/ , n is 07/ 2nbh + 7ia 2 \ 
= 6(cn+2^)+^a 2 =6cfnH ^ J. 




302 



APPENDIX 



The second term in the parentheses is constant (indepencU 
ent of n) and should be engraved on, or otherwise supplied 
with, the instrument. This number is then to be looked upon 
as a correction to n when the planimeter makes a complete 
revolution. 27ibh + iza 2 is evidently the area of the circle 
described by B when n remains =0, i.e., when the instru- 
ment is set so that the wheel slides without rolling, or when 
the perpendicular from C on AB passes through the wheel. 

By sliding the bar AB through a sleeve to which the 
hinge and the wheel are attached, its length may be altered 
and the instrument adapted to different units. Thus if the 
circumference of the wheel =c centimetres, and b is taken 
= 100/c, bcn = 100n, and hence the area is found in square 
centimetres by multiplying n by 100. Similarly if the 
circumference of the wheel is c inches and b is taken = 10/ c, 
the area ben/ = 10n square inches. 

6. As we proceed from B to C by way of P, Fig. 146, 

I* 
x changes from OD to OE, and y dx is the area DBPCE; 

but if we proceed from C to B by way of P r , each element 
of area such as y dx is negative since dx is negative, and 



hence 



y dx is the area CEDB, but is negative. Hence if 



we sum the elements such as y dx in the order of proceeding 
clockwise round the curve, the result =DBPCE - DBP'CE = 
BPCP', the area of the curve. Let A = this area, M = the 
sum of the moments of the elements of the area w T ith respect 
to OX, 7= the momerrt of inertia of the area with respect 
to OX. Then 



A = 



y dx, M 



' y__ 1 

y ax . Q ~ o 



y 2 dx, I = y dx . 



W 



1 
3 



y s dx. 



7. Amsler's mechanical integrator. In this instrument 
one end of a sweeping bar FP traces a closed curve, while 
the other end is constrained to describe a straight line OX. 
Hence this part of the instrument is virtually a planimeter, 
and the area A = bcin, w r here b is the length of FP, c x the cir- 



MECHANICAL INTEGRATION. 



303 



cumference of the wheel W x which FP carries, and n x the 
change of reading of this wheel when the circuit of the curve 
has been made. FP also turns two arcs of centre F and 
radii 2a and 3a, which turn circles each of radius a, the circles 
carrying wheels W 2 and W s . The three wheels roll simul- 
taneously on the plane containing the diagram to be inte- 
grated. When the axis of W ± makes an angle with OX 




Fig. 146. 

the axes of W 2 and W 3 make angles \iz— 20 and 30, respect- 
ively, with the same line. 

For y substitute b sin 0. Then 

A = b sin dx, M =>\b 2 |"sin 2 dx, I = J6 3 [sin 3 dx. 

Now 2 sin 0=1 -cos 20 = 1 -sin (^-20), 

and sin 30 = 3 sin — 4 sin 3 0, or sin 3 = f sin — J sin 30. 



.-. M=W 



[1-sin (fr-2d)]dx= -\b sin {\n-2d)dx J 



since dx=0 for the complete circuit of the curve. 



Also, 



1=W 
= i& 3 



(fsin0-isin30)dx 



sin ddx—ihb 2 



sin 30 dx. 



304 APPENDIX. 

But A = bciUi f .*. sin 6 dx=cini, i.e., when the axis of a 
wheel makes an angle 6 with OX, sin 6 dx=C\U\. But the 

■ 

axes of the other wheels make angles \tz — 2d and 3d with OX. 
/. sin (\n — 2d)dx = c 2 n 2 , and sin 30 dx C3713. 

.\ A =bcini, M=lb 2 c 2 n 2 , I =J6 3 cini — Y2^ Sc s n s- 

In the instrument under discussion the maker has taken 
6=2 decimetres, ci= J dec, c 2 =cs =f dec. 

.-. A=n 1} M = fn 2 , /=m-|n 3 . 

The height y of the centre of gravity of the area above 
0X = M/A, and the moment of inertia with respect to an 
axis through the centre of gravity and parallel to 
0X = I-AP = I-M 2 /A. 

Results may be changed from decimetres to inches by 
multiplying y by a, A by a 2 , M by a 3 , and I by a 4 , where 
a=3*937, the number of inches in a decimetre; a 2 = 15*500, 
a 3 = 61'023, a 4 = 240'290. 






MISCELLANEOUS EXAMPLES. 

1. Prove Leibnitz 's theorem for the nth differential of a product: 

71(71 — 1 ) 
d n (uv) = (d n u)v + nd n ~ 1 u dv + — — — d n ~ 2 ud 2 v + . . ,+ud n v. 

[By induction from d(uv) =v du+u dv.] 

2. If y 2 =a 2 +2xy, d 2 y/dx 2 =a 2 /(y — x) 3 and d 2 x/dy 2 = —a 2 /y*. 



3. The maximum value of 



(1) X =1-202. 



4. Show that the turning points of the curve y = sin x- 1 are 
where x=2/(nn), n being any odd integer. (The number when 
x = from any assigned value is therefore infinite.) 

5. Given the volume of a right circular cylinder, show that the 
surface is a minimum when the altitude =the diameter. 

6. The height of the greatest rectangle which can be inscribed 
in a given right segment of a parabola is two-thirds of the height 
of the segment. 

7. Find the area of the rectangle circumscribing the loops of 
the curve ay 3 -Sax 2 y=x 4 (Fig. 36). Arts. 9a 2 . 

8. 



4 sin 4 0d0=*O45, 9. |tan 4 d0=*119. 



JO 



f 4 f °° dx 

10. sec 4 0dfl=H. 11. -7— -i=-215. 

Jo Ji x +x 

10 I" 1 dx f°°__^_ _3? 

1Z ' ] l+x+x 2 ~ W ^ l6 ' J (i+*yi6- 

m M [ a x 2 dx , - „„ [s dx 

14. ~m~ w*='174:. 15. =T317. 

J (x 2 +a 2 )* J cos a; 



n 



.„ f 3 dx . „ n ^ „„ [3 dx 

16. — r =l'732. 17. — r =2*391. 

J cos 2 # J cos 3 a* 



305 



306 



APPENDIX. 



18. 
20. 

22. 



it 

3~ dx 







cos 4 :r 



=3-464. 



2 



sec x dx='522. 



o 



2 dx 2 

(1 +cos x) 2 3 ' 



7T 

-.. f2l — cos 3 # 7 
24. — ^—dx=2. 
J sm 2 z 



26. 

28. 
30. 

32. 

34. 
36. 

38. 
39. 
40. 



dx 



a^{x — a){b — x) 



= n. 



'2 dx 



2 + cos x 



='604. 



o 

1 x dx 

(x + l)(x 2 + l) 



x%(l —x)%dx=-^^. 



'cos \Q 



128* 



sin 



dd = X 



(V)- 



foo 



6 _a:r cos mo: dx 



o 

* / x 2 



a 



19. 
21. 

23. 

25. 
27. 
29. 



•2 



sin 2 x^=-002. 



ri 



e^cos x dx = 1*378. 



o 

3~ eta* 



* sin 2x 
1 



='275. 



dtf 



^sin 2 ^ cos 6 
T 



=•695. 



2+ cos x 



dx 



='219. 31. 
33. 



sin £ + cos a; 



= 1*813. 



--=•810. 



x 2 (l-^)^='152. 



a 



a — x 



S 







:- 



X ) cos wx eta 



a 2 +m 2 
1 



35. 
.37. 



x 



dd 



dx=^7za. 



g + ^g-m+logCOBd. 



sec 

1 dx 



a 







# 2 +2a; cos a + 1 2 sin a' 



m 



2* 



d# 



1 , ft n\ 

=-rtan- 1 1— tan ) . 
ab \a J 



a 2 cos 2 + b 2 sin 2 # 

dx 2 (x + a)% — (x + b)* 



Vx + a + Vx+b 3 a-b 



n 



n 



" ,+ 2ttV 



"* dx 7T 

r+7 2= 4" 







III/ IV 

41 * £n = cc \ n 2 + l 2 + 7l 2 +2 2 + 

Let l/n=dx. 

42. The area of the evolute of the ellipse =— : — n. 

8 ab 



MISCELLANEOUS EXAMPLES. 307 

43. A cycloid revolves about its axis of symmetry. Show that 
the volume of the solid is 7ra 3 (%x 2 — f) and that the convex surface 
is 87ra 2 (7r-|). 

44. A hemispherical bowl of 1 ft. radius is filled with water 
which then runs out of an orifice at the bottom \ sq. in. in sec- 
tional area. Find the time of emptying, assuming that the 
velocity at the orifice = ^2gx, where x is the height of the surface 
of the water above the orifice. Arts. 1 m. 46 s. 

45. Find the centre of gravity of the area between the curve 
(x/a)* + (y/b)% = 1 and the axes. Ans. ~x =ia } y =\b. 

46. Prove that X(x) =x +— +— + . . . 

mm -, i O n= i°cosn7r Qosnx 

47. a;sina; = l-i cos x=2 I — , for \ — n, n\. 

n=2 U 2 -l 

i . J 1 ^ ( - l) n n sin nx . . 

48. x cos x= —i sm x+2 2 — -, for J — n, n[. 

n=2 % 1 

49. x 3 =— +— 2 — cos?i7rH — 4 (1— cos W7r) cos no:, for [0, iz\. 

4 n n =i Ln 2 n J 

1 1 1 7T 4 1 1 1 ^ 

Hence prove that -^ + - 4 +j A +. . . =— , — +— +— + . . . =— . 

50. Find (1) the area, and (2) the length, of one loop of the 
curve r n =a n cos nd. 

Wi+I\ r(l\ 

■ m 2 ifp <2) ~Ta^r\ 

\nl \2n 2/ 

51. Find the area in the first quadrant between the axes and 

«—(!)"+(!)"-• t ['•(i+i)]' 

Ans. -; — — ab. 

r(i+±) 

Show that the area = ab when n is infinite. 

52. Find the area of x 3 +y 3 =a 3 in the first quadrant, and the 
whole area of x A + y 4 =a\ Ans. 0*883a 2 , 3*708a 2 . 

53. Find the area of one oval of the curve y 2 =a 2 sin (x/b). 

Ans. 4*792a&, 

2" dd 



54. Show that 



Vsin 6 dd . 
o 



o Vsin 6 



7Z. 



308 APPENDIX. 

55. The curve r m cos md=a m rolls on a straight line. To find 
the differential equation of the locus of the polar origin. 

Let the straight line be taken as z-axis, and the polar origin 
be at (x, y). Then r=the normal at (x, y), and 2/=the perpen- 
dicular on the tangent at (0, r). Hence 



u 



yVl +p*' y 



1 o fdu\ 2 
, —=u +\T d ) , a m u m =cosmd. 



Eliminate u and 6 and show that (1 +p 2 ) 1 ~ m = {y/a) 2m . 

56. A parabola rolls on a straight line. Show that the focus 
describes a catenary. 

57. Find the centre of gravity of the arc of the quadrant of 
an ellipse (semi-axes a and b, eccentricity e). 



Ans. 5-| (l +^(1 -e 2 ) log ~~j /E(e, fr), 
y =- (Vl-e 2 -f-sin-^j /E(e, in). 



58. An ellipse (eccentricity e) and a circle have equal areas. 

Find the ratio of their circumferences. Ans. — — L -~ . 

7r(l — e 2 p 

59. Find the curves which make an angle a with the curves 

r n = a n cos n () j r n cos n Q = a n t 

Ans. r n = c n cos (nd + a), r n cos (nO — a) = c n . 

™ ^. dx dy . ^ dx dy rt . , .- 

60. Given — -~ +w = sm 2J, — +-/ +# = 0, show that 

dt dt * ' dt dt 

t t 



x = ce 2 + c x e 2 — f cos 2£, 



t t 



y = -(V2 + l)ce V2 + (V2-l)cx6 V2 +| sin 2* +f cos 2*. 



TABLES. 

PAGE 

1. POWERS, NAPIERIAN LOGARITHMS, ETC 310 

2. CIRCULAR FUNCTIONS, 1 312 

3. CIRCULAR FUNCTIONS, II 313 

4. HYPERBOLIC FUNCTIONS 314 

5. LAMBDA FUNCTION 315 

6. GAMMA FUNCTION 315 

7. FIRST ELLIPTIC INTEGRAL 316 

8. SECOND ELLIPTIC INTEGRAL 3*6 

309 



310 



TABLES. 





1 


. POWERS, NAPIERIAN 


LOGARITHMS, 


ETC 


1 


e(io^) 


X 


x~ l 


X 2 


X 3 


X 1 


(io*)£ 


X 5 


log e # 


log 


0*1 


IO 





■OI 





'OOI 





316 


1 


000 


0-464 


-2-303 


o-oco 


0*2 


5 


ooo 





•04 





• 008 





447 


1 


414 





585 


— I 


609 





6 93 


o*3 


3 


333 





■09 





'027 





548 


1 


732 





696 


— I 


204 


I 


099 


0*4 


2 


500 





l6 





•064 





632 


2 


000 





737 


— O 


916 


I 


386 


o'5 


2 


000 





•25 





125 





707 


2 


236 





794 


— O 


6 93 


I 


609 


o*6 


I 


667 





36 





216 


o- 


775 


2 


449 





843 


— p- 


5 J i 


I' 


792 


o*7 


I 


429 





49 





343 


O' 


837 


2 


646 





888 


— O- 


357 


I' 


946 


o-8 


I- 


250 





64 





512 





894 


2 


828 





928 


— O' 


223 


2- 


079 


o-9 


I- 


III 





81 


0< 


729 


O' 


949 


3 


000 





965 


— 0- 


i°5 


2- 


197 


10 


I' 


000 


1 


00 


I 


000 


I ■ 


000 


3 


162 


1 


000 


O' 


000 


2« 


3°3 


ri 


0- 


909 


i- 


21 


I' 


33* 


I' 


049 


3" 


317 


I- 


032 


O' 


095 


2- 


398 


1*2 


O- 


833 


1 


44 


I< 


728 


I' 


°95 


3' 


464 


1 ■ 


063 


o- 


182 


2- 


485 


i'3 


o< 


769 


1 


69 


2- 


197 


I- 


140 


3" 


606 


1 • 


091 





262 


2- 


5 6 5 


i*4 


o 


714 


1 


96 


2 


744 


I' 


183 


3' 


742 


i- 


119 


o- 


33b 


2 


639 


i'5 





667 


2 


25 


3 


375 


I 


225 


3' 


873 


1 


!45 


O' 


405 


2- 


708 


r6 





625 


2 


56 


4 


096 


I 


265 


4' 


000 


i' 


170 


O' 


470 


2- 


773 


i'7 


o 


588 


2 


89 


4 


9*3 


I 


3°4 


4' 


123 


i- 


J 93 


o- 


53 1 


2 


^33 


r8 


O' 


556 


3 


24 


5 


832 


I« 


342 


4' 


243 


i« 


216 


o< 


588 


2 


890 


rg 


o 


526 


3 


61 


6 


859 


I' 


378 


4' 


359 


i- 


239 


o- 


642 


2 


944 


2*0 


o 


500 


4 


•00 


8 


-ooo 


I 


414 


4' 


472 


i' 


260 


0- 


6 93 


2 


996 


2*1 


o 


•476 


4 


-41 


9 


261 


I 


449 


4' 


583 


I- 


281 


o« 


742 


3 


045 


2*2 


o 


•455 


4 


84 


10 


648 


I 


483 


4' 


690 


I- 


301 


o< 


788 


3 


091 


2 # 3 


o 


•435 


5 


•29 


12 


167 


I 


517 


4' 


796 


I- 


320 


O' 


833 


3 


!35 


2-4 


o 


•417 


5 


76 


13 


824 


I 


549 


4' 


899 


1 ■ 


339 


0' 


875 


3' 


178 


2*5 


o 


400 


6 


,2 5 


15 


.625 


I 


58i 


5 


000 


i- 


357 


o- 


916 


3 


219 


2*6 


o 


385 


6 


76 


17 


576 


I 


612 


5 


099 


I- 


375 


o- 


956 


3" 


258 


2'7 





37° 




•29 


*9 


.683 


I 


643 


5' 


196 


1 


392 


O' 


993 


3 


296 


2-8 


o 


357 


7 


84 


21 


'952 


I 


6 73 


5 


292 


1 


41c 


I 


030 


3 


33 2 


2*9 


o 


345 


8 


4i 


24 


389 


I 


7°3 


5 


385 


1 


426 


I 


065 


3 


367 


3'0 


o 


333 


9 


00 


27 


000 


I 


■732 


5 


'477 


1 


442 


I 


099 


3 


401 


3'i 





3 2 3 


9 


61 


29 


79 1 


I 


761 


5 


568 


1 


458 


I 


131 


3 


434 


3- 2 





'3*3 


10 


■24 


32 


•768 


I 


789 


5 


657 


1 


•474 


I 


163 


3 


466 


3*3 


o 


'3°3 


10 


.89 


35 


•937 


I 


•817 


5 


•745 


1 


■489 


I 


194 


3 


'497 


3 # 4 


o 


294 


11 


•56 


39 


•3°4 


I 


■844 


5 


■831 


1 


•5°4 


I 


224 


3 


526 


3*5 


o 


286 


12 


■ 2C 


42 


•875 


I 


.871 


5 


•9l6 


1 


.518 


I 


■253 


3 


'555 


3'6 


o 


278 


12 


.96 


46 


.656 


I 


•897 


6 


•OOO 


1 


•533 


I 


•281 


3 


584 


3*7 


o 


-270 


I 3 


.69 


50 


'653 


I 


.924 


6 


■08 3 


1 


'547 


I 


.308 


3 


611 


3'8 


o 


-263 


14 


.44 


54 


•£72 


I 


•949 


6 


' 164 


1 


■561 


I 


'335 


3 


638 


3*9 


o 


256 


15 


21 


59 


■3i9 


I 


•975 


6 


•245 


1 


■574 


I 


•361 


3 


664 


4*o 


o 


250 


16 


OC 


64 


'OOO 


2 


•00c 


6 


'3 2 5 


1 


■587 


I 


■386 


3 


■689 


4'i 


o 


244 


16 


8l 


68 


■921 


2 


-025 


6 


403 


1 


•601 


I 


•411 


3 


714 


4*2 


o 


238 


17 


-64 


74 


• 088 


2 


.049 


6 


481 


1 


•613 


I 


'435 


3 


738 


4*3 


o 


2 33 


18 


49 


79 


'5°7 


2 


•074 


6 


•557 


1 


•626 


I 


•459 


3 


761 


4*4 


o 


227 


19 


36 


85 


•184 


2 


'098 


6 


^33 


1 


.639 


I 


•482 


3 


•784 


4*5 


o 


222 


20 


25 


9i 


I2 5 


2 


121 


6 


708 


1 


651 


I 


'5°4 


3 


•807 


4'6 


o« 


217 


21' 


16 


97 


336 


2 


!45 


6 


782 


1 


663 


I 


526 


3 


829 


4*7 


o- 


213 


22 


09 


103 


823 


2 


168 


6 


856 


1 


■675 


I 


548 


3 


850 


4*8 


o- 


208 


2 3 


04 


no 


592 


2 


191 


6 


928 


1 


687 


I 


569 


3 


■871 


4*9 


o- 


204 


24 


01 


117 


649 


2 


214 


7 


000 


1 


699 


I 


589 


3 


892 


5*o 


O • 200 


25-OC 


125-000 


2- 236 


7-071 


1 • 710 


I • 609 


3.912 



e = 



2*71828, log c 10=2'30259, log 10 e = 0'4342y. 



TABLES. 



311 





1 


. POWERS, NAPIERIAN 


LOGARITHMS 


ETC. 




X 


XT 1 


X 2 


X 3 


1 

X 2 


(io#)* 


1 

X* 


loge * 


loge(iox) 


5'o 


O 


• 200 


2 5 


•00 


125 


•000 


2 


•236 


7 


•071 


1 


• 710 


I 


609 


3 


912 


5'i 


O 


• I96 


26 


■01 


132 


■651 


2 


•258 


7 


•141 


1 


•721 


I 


'629 


3 


93 2 


5*2 


O 


• I92 


27 


•04 


140 


•608 


2 


.280 


7 


•211 


1 


•732 


I 


649 


3 


95i 


5*3 


O 


•189 


28 


.09 


148 


■877 


2 


.302 


7 


•280 


1 


"744 


I 


668 


3 


970 


5*4 


O 


.185 


29 


•16 


157 


•464 


2 


.324 


7 


■348 


1 


'754 


I 


686 


3 


989 


5*5 


o 


•l82 


30 


•25 


166 


■375 


2 


•345 


7 


416 


1 


•765 


I 


7°5 


4 


007 


5*6 


o 


•179 


3 1 


■36 


175 


.616 


2 


.366 


7 


•483 


1 


776 


I 


•723 


4 


025 


5*7 


o 


•175 


3 2 


•49 


185 


• J 93 


2 


•387 


7 


•55o 


1 


•786 


I 


.740 


4 


043 


5-8 


o 


• 172 


33 


.64 


195 


- 112 


2 


•408 


7 


■616 


1 


•797 


I 


758 


4 


060 


5' 9 


o 


• 169 


34 


■81 


205 


'379 


2 


•429 


7 


•681 


1 


■807 


I 


•775 


4 


078 


6-o 


o 


• 167 


36 


■00 


216 


■ooo 


2 


"449 


7 


.746 


1 


.817 


I 


.792 


4 


094 


6- 1 


o 


• 164 


37 


•21 


226 


•981 


2 


■470 


7 


810 


1 


-827 


I 


808 


4 


in 


6'2 


o 


•161 


38 


•44 


238 


•328 


2 


•49° 


7 


•874 


1 


•837 


I 


•825 


4 


127 


6*3 


o 


•159 


39 


.69 


250 


047 


2 


.510 


7 


•937 


1 


•847 


I 


841 


4 


•143 


6*4 


o 


.156 


40 


.96 


262 


•144 


2 


•53o 


8 


•000 


1 


•857 


I 


856 


4 


•159 


6*5 


o 


•154 


42 


•25 


274 


•625 


2 


•55o 


8 


•062 


1 


•866 


I 


•872 


4 


•174 


6-6 


o 


•i5 2 


43 


•56 


287 


•496 


2 


•5 6 9 


8 


■ 124 


1 


•876 


I 


•887 


4 


• 190 


6° 7 


o 


•149 


44 


.89 


300 


■763 


2 


•588 


8 


■18s 


1 


•885 


I 


902 


4 


•205 


6-8 


o 


•147 


46 


•24 


3i4 


•432 


2 


•608 


8 


•246 


1 


•895 


I 


917 


4 


■ 220 


6*9 


o 


•145 


47 


•61 


328 


•5°9 


2 


•627 


8 


'3°7 


1 


•904 


I 


•932 


4 


•234 


7*o 


o 


•143 


49 


•00 


343 


•000 


2 


■646 


8 


•367 


1 


•9 X 3 


I 


.946 


4 


-248 


7-i 


o 


•141 


5° 


•41 


357 


.911 


2 


.665 


8 


•426 


1 


•922 


I 


•960 


4 


•263 


7-2 


o 


•!39 


5 1 


.84 


373 


•248 


2 


•683 


8 


•485 


1 


■93 1 


I 


•974 


4 


•277 


7'3 


o 


•137 


53 


•29 


389 


•017 


2 


• 702 


8 


•544 


1 


.940 


I 


•988 


4 


•290 


7*4 


o 


•135 


54 


.76 


405 


•224 


2 


• 720 


8 


•602 


1 


•949 


2 


•001 


4 


•3°4 


7*5 


o 


•!33 


56 


' 2 5 


421 


■875 


2 


•739 


8 


-66o 


1 


•957 


2 


•015 


4 


•317 


7*6 


o 


■132 


57 


.76 


438 


.976 


2 


•757 


8 


•718 


1 


.966 


2 


•028 


4 


•33i 


7*7 


o 


•130 


59 


■29 


45 6 


'533 


2 


•775 


8 


■775 


1 


•975 


2 


■041 


4 


•344 


7'8 


o 


•128 


60 


.84 


474 


•552 


2 


•793 


8 


•832 


1 


•983 


2 


•054 


4 


•357 


7*9 


o 


• 127 


62 


■41 


493 


■°39 


2 


■811 


8 


•888 


1 


•992 


2 


•067 


4 


•369 


8-o 


o 


■125 


64 


•00 


512 


•000 


2 


•828 


8 


•944 


2 


•000 


2 


•079 


4 


.382 


8-i 


o 


■123 


65 


61 


53i 


•441 


2 


•846 


9 


•00c 


2 


• 008 


2 


•092 


4 


■394 


8*2 


o 


■ 122 


67 


■24 


55i 


.368 


2 


•864 


9 


•°55 


2 


•017 


2 


• 104 


4 


•407 


8-3 


o 


■ 120 


68 


.89 


57i 


•787 


2 


•881 


9 


• no 


2 


•025 


2 


•116 


4 


.419 


8-4 


o 


•119 


70 


56 


592 


.704 


2 


•898 


9 


• 165 


2 


'°33 


2 


•128 


4 


•43i 


8*5 


o 


118 


72 


•25 


614 


■125 


2 


■9i5 


9 


•220 


2 


•041 


2 


140 


4 


•443 


8-6 


o 


116 


73 


96 


636 


■056 


2 


•933 


9 


•274 


2 


•049 


2 


•152 


4 


•454 


8-7 


o 


115 


75 


69 


658 


■5°3 


2 


•95° 


9 


•327 


2 


•°57 


2 


163 


4 


•466 


8-8 


o 


114 


77' 


44 


681 


•472 


2 


•966 


9 


•381 


2 


•065 


2 


.175 


4 


•477 


8-9 


o- 


112 


79' 


21 


704 


.969 


2 


983 


9 


•434 


2 


•072 


2 


186 


4 


•489 


9-o 


o- 


in 


81 ■ 


00 


729 


■ooo 


3 


000 


9 


487 


2 


•080 


2 


197 


4 


•500 


9-i 


o« 


no 


82- 


81 


753 


571 


3 


017 


9 


539 


2 


088 


2 


208 


4 


5ii 


9*2 


o- 


109 


84- 


64 


778 


688 


3 


°33 


9 


592 


2 


095 


2 


219 


4 


522 


9*3 


o- 


108 


86- 


49 


804 


357 


3 


050 


9 


644 


2 


103 


2 


230 


4 


533 


9*4 


o- 


106 


88- 


36 


830 


584 


3 


066 


9' 


695 


2 


no 


2- 


241 


4' 


543 


9' 5 


o- 


105 


90- 


25 


857 


375 


3 


082 


9' 


747 


2 


118 


2- 


25 1 


4' 


554 


9-6 


o- 


104 


92. 


16 


884- 


736 


3 


098 


9' 


798 


2 


125 


2« 


262 


4" 


5 6 4 


9*7 


o- 


103 


94. 


09 


912- 


673 


3' 


114 


9" 


849 


2- 


*33 


2- 


272 


4" 


575 


9'8 


O' 


102 


96. 


04 


941- 


192 


3" 


130 


9' 


899 


2- 


140 


2- 


282 


4' 


585 


9*9 


o« 


IOI 


98. 


01 


970- 


299 


3' 


146 


9" 


95o 


2« 


147 


2- 


2Q3 


d- 


<qk 



log 10 z=0*43429 log e x, log e z-2'30259 log 10 x. 



312 



TABLES 











2. 


CIRCULAR 


FUNCTIONS 


I. 










d 


6 


sin 6 


cosec 6 


tan 6 


cot 6 


sec 6 


cos 6 




" 


Degrees 


Radians 





















O 


000 





•000 


00 





-ooo 


00 


1 


•000 


1 


•000 


i-57i 


90 


I 


O 


•017 





•017 


57 


•299 





•017 


57 


•290 


1 


•000 


1 


000 


i" 


553 


89 


2 


O 


'°35 





•035 


28 


•654 





035 


28 


■636 


1 


•001 





999 


I- 


536 


88 


3 


O 


•052 





■052 


r 9 


107 





052 


J 9 


081 


1 


'OOI 





•999 


i- 


5i8 


87 


4 


O 


■070 





►070 


14 


336 


o< 


070 


14 


301 


1 


'O02 





998 


i« 


501 


86 


5 


O 


•087 





•087 


11 


•474 


o< 


087 


11 


•43° 


1 


004 





996 


1 ■ 


484 


85 


6 


O 


105 





105 


9 


567 


o- 


105 


9 


5i4 


1 


006 





995 


1 


466 


84 


7 


O 


122 





122 


8 


206 


o- 


123 


8 


144 


1 


008 





993 


1 


449 


83 


8 


O 


140 


0- 


139 


7 


185 


O' 


141 


7 


"5 


1 


010 





990 


1 


43 1 


82 


9 


O 


157 


0- 


156 


6 


392 


o- 


158 


6 


3i4 


1 


012 





988 


1 • 


414 


81 


10 


0< 


175 


o« 


174 


5 


759 


O' 


176 


5 


671 


1 


01 5 





98^ 


1 


39 6 


80 


ii 


o- 


192 


o- 


IQI 


5 


241 


o< 


194 


5 


T 45 


I- 


019 





982 


1 


379 


79 


12 





209 


0- 


208 


4 


810 


o- 


213 


4 


7°5 


i" 


022 





978 


1 ■ 


361 


78 


13 


o« 


227 





225 


4 


445 


O' 


231 


4 


33* 


1 


026 





974 


I- 


344 


77 


14 





244 





242 


4 


1 34 





249 


4 


on 


i« 


031 





970 


1 ■ 


326 


76 


15 





262 





259 


3 


864 


o- 


268 


3 


732 


1 


°35 


o- 


966 


I; 


309 


75 


16 





279 





276 


3 


628 





287 


3 


487 


1 • 


040 


O' 


961 


I ■ 


292 


74 


17 


o- 


297 





292 


3 


420 





306 


3 


271 


i- 


046 


o< 


956 


I- 


274 


73 


18 





3*4 





3°9 


3' 


2-36 


0- 


325 


3 


078 


1 ■ 


°5i 


o- 


95i 


I- 


257 


72 


19 





33 2 





326 


3 


072 


o- 


344 


2" 


904 


i« 


058 


o- 


946 


I- 


239 


7i 


20 





340 





342 


2- 


924 


o« 


364 


2 


747 


i- 


064 


O" 


940 


I« 


222 


70 


21 





367 





358 


2 


790 


o< 


384 


2 


605 


1 


071 


o- 


934 


I« 


204\ 


• 69 


22 





384 





375 


2 


669 





404 


2 


475 


i- 


079 


o« 


927 


I • 


i?7 


68 


23 





•401 





39 1 


2 


559 





424 


2 


356 


i- 


086 


o« 


921 


I- 


169 


67 


24 





419 





•407 


2 


459 





445 


2 


246 


I- 


095 


O" 


914 


I« 


152 


66 


25 





43 6 





423 


2 


3 66 





466 


2 


145 


i- 


103 


O' 


906 


« I- 


134 


65 


26 





454 





438 


2 


281 





488 


2 


050 


I- 


"3 





899 


I« 


117 


64 


27 





471 





'454 


2 


203 





.510 


I 


963 


I- 


122 





891 


I 


IOO 


63 


28 





489 





469 


2< 


130 





•532 


I 


881 


1 


133 





883 


I 


082 


62 


29 





506 





485 


2 


063 





•554 


I 


804 


1 


J43 





875 


I 


065 


61 


30 





5 2 4 





500 


2 


000 





•577 


I 


732 


1 


x 55 





866 


I 


-047 


60 


31 





54i 





515 


I 


942 





•601 


I 


664 


1 


•167 





•857 


I 


•030 


59 


32 





559 





53o 


I 


887 





625 


I 


■600 


1 


•179 





848 


I 


•OI2 


58 


33 





576 





■545 


I 


836 





.649 


I 


•54o 


1 


■ 192 





•839 


O 


"995 


57 


34 





593 





•559 


I 


788 





•675 


I 


•483 


1 


•206 





■829 


O 


'977 


56 


35 





611 





•574 


I 


743 





•700 


I 


•428 


1 


■221 





•819 


O 


•960 


55 


36 





628 





•588 


I 


•701 





.727 


I 


'376 


1 


•236 





•809 


O 


■942 


54 


37 





646 





602 


I 


662 





754 


I 


•327 


1 


•252 





•799 





•925 


53 


38 





.663 





616 


I 


624 





.781 


I 


•280 


1 


•269 





-788 


O 


»908 


52 


39 





681 





'629 


I 


589 





810 


I 


•235 


1 


•287 





•777 


O 


•89O 


5i 


40 





698 





643 


I 


556 





839 


I 


192 


1 


3°5 





•766 


O 


•873 


50 


4i 





716 





6^6 


I 


524 





869 


I 


*5° 


1 


325 





•755 


O 


•855 


49 


42 





733 





669 


I 


494 





900 


I ■ 


in 


i« 


346 





•743 


O 


838 


48 


43 





75° 





682 


I" 


466 





933 


I ■ 


072 


i« 


367 





73 1 


O 


82C 


47 


44 





768 





695 


I< 


440 





966 


I- 


036 


i- 


39° 





719 





80.3 


46 


45 





785 


o< 


707 


I- 


414 


I 


000 


I • 


000 


i« 


414 


0*707 


0.78;! 


' 45 






cos 


sec d 


cot 6 


tan d 


cosec 6 


sin 6 


6 1 



































Pa 


rliars 


Degrees 



lrdn. = 57°*29578,l°=001745idn.,lrdn. = 206265 ,, ,l ,, = 000004848rdn. 



TABLES. 



313 



3. CIRCULAR FUNCTIONS, II. 



6 


6 


sin 6 


cosec 6 


tan 


cot 


sec d 


cos 6 


Jladians 


Degrees 














0*00 


o-oo 


o-ooo 


00 


o-ooo 


00 


i- 000 


I'OOO 


O'OI 


o-57 


O-OIO 


100 • 002 


O-OIO 


99-997 


i- 000 


I -ooo 


0*02 


i' iS 


0-020 


50-003 


0-020 


49-993 


i- 000 


I -ooo 


o-03 


1-72 


0-030 


33-338 


0-030 


33'3 2 3 


1 -ooo 


1 -ooo 


0*04 


2-29 


0-040 


25-007 


0-040 


24-987 


I -OOI 


o-999 


0-05 


2-86 


0-050 


20 • 008 


0-050 


19-983 


I -OOI 


0-999 


o*o6 


3'44 


o-o6o 


16-677 


o-ooo 


16-647 


I -002 


0-998 


0-07 


4-01 


0-070 


14-297 


0-070 


14-263 


I -002 


0-998 


o-o8 


4-58 


0-080 


12-513 


0-080 


12-473 


1-003 


o-997 


o-og 


5.16 


0-090 


ii- 126 


0-090 


11 -081 


1-004 


0-996 


o-i 


5-73 


o« 100 


10-017 


o ; 100 


9.967 


1-005 


o-995 


0*2 


11-46 


0-199 


5- °33 


0-203 


4*933 


I -020 


0-980 


o-3 


17-19 


0-296 


3-3 8 4 


0-309 


3- 2 32 


1-047 


o-955 


o-4 


22-92 


0.389 


2.568 


0-423 


2-3°5 


I- 086 


0-921 


o-5 


28-65 


0-479 


2- 086 


0-546 


1.830 


!• I4O 


0-878 


o*6 


34-38 


o-565 


I- 771 


0-684 


1-462 


I-2I2 


0-825 


o*7 


40-11 


0-644 


1-552 


0-842 


1-187 


I.307 


0-765 


o-8 


45-84 


0-717 


1*394 


1-030 


0.971 


1*435 


0-697 


o-g 


5*'57 


0-783 


1-277 


I • 260 


0-794 


I -609 


0-622 


I"0 


57*3o 


0-842 


i- 188 


1-558 


0-642 


I-8 5 I 


0-540 


I* i 


63 '°3 


0-891 


1 • 122 


1-965 


0-509 


2-205 


o-453 


1-2 


68.75 


0-932 


1-073 


2-571 


0.389 


2- 760 


0-362 


i'3 


74.48 


0-964 


1-038 


3-602 


0-278 


3-738 


0-268 


i - 4 


80 • 21 


0-985 


1-015 


5-798 


o- 172 


5-884 


o- 170 


i-5 


85-94 


0-997 


1-003 


14- 101 


0-071 


14.136 


0-071 


\k 


90*00 


I'OOO 


I 'OOO 


00 


O'OOO 


00 


o-ooo 



7z = 3*14159, 
7T- 1 =0*31831, 



7r 2 = 9'86960, 

7T- 2 =0*10132, 



VW T77245, 
V^F 1 =0*56419. 



314 



TABLES. 









4. HYPERBOLIC FUNCTIONS. 










X 


e x 


e —x 


sinh x 


cosha; 


tanh x 


coth x 


sech# 


cosec h x 


O'O 


I • 


000 


I' 000 


i- 


000 


1 • 


000 


o- 


000 


00 


1 ■ 


000 


00 


o-i 


I • 


105 


0.905 


o- 


100 


1 • 


005 


0- 


100 


IO« 


°35 





995 


9 


985 


0*2 


I • 


221 


0-819 


o- 


201 


i' 


020 





197 


5 


067 





980 


4' 


975 


0'3 


I- 


35° 


0.741 


o- 


3°5 


i- 


045 


o- 


291 


3 


433 





957 


3" 


284 


o-4 


I • 


49 2 


0-670 


O' 


411 


1 ■ 


081 





380 


2 


632 





925 


2" 


433 


o'5 


I ■ 


649 


0-607 


o- 


521 


i« 


128 


o< 


462 


2 


164 





887 


I 


919 


o*6 


I ' 


822 


o-549 





637 


1 


185 





537 


1 


862 





844 


I 


57o 


o-7 


2- 


014 


0-497 


O" 


759 


I- 


2 55 


o- 


604 


1 


655 





796 


I- 


3i8 


o-8 


2 


226 


0-449 





888 


1 


337 





664 


1 


506 





748 


I' 


126 


o*9 


2 


460 


0-407 


I 


027 


1 


433 





716 


1 


39° 





698 


O 


974 


1*0 


2 


•718 


0-368 


I 


175 


1 


543 





762 


1 


3i3 





648 


O 


851 


i- 1 


3 


004 


°'333 


I 


33 6 


1 


669 


o- 


800 


1 


249 





599 


O 


749 


1*2 


3 


■320 


0-301 


I 


5°9 


1 


811 





834 


1 


200 





552 


O 


662 


i*3 


3 


•669 


0-273 


I 


698 


1 


971 





862 


1 


161 





507 




589 


i'4 


4 


•o55 


0-247 


I 


9°4 


2 


J5 1 





885 


1 


129 





465 


O 


525 


i-5 


4 


•482 


0-223 


2 


• 129 


2 


35 2 





9°5 


1 


•105 





•425 


O 


47° 


i-6 


4 


■953 


0-202 


2 


■376 


2 


577 





922 


1 


085 





388 


O 


421 


i-7 


5 


•474 


0-l8 3 


2 


•646 


2 


■828 





■935 


1 


069 





354 


O 


378 


r8 


6 


•050 


0-165 


2 


.942 


3 


■ 107 





■947 


1 


056 





322 


O 


340 


i-9 


6 


•686 


0-I50 


3 


•268 


3 


■418 





95 6 


1 


046 





293 


O 


306 


2'0 


7 


•389 


o-!35 


3 


•627 


3 


■762 





.964 


1 


o37 





•266 


O 


276 


2*1 


8 


•166 


O- 122 


4 


•022 


4 


•144 





•970 


1 


031 





241 


O 


249 


2*2 


9 


.025 


O- III 


4 


•457 


4 


.568 





•976 


1 


•025 





219 


O 


224 


2-3 


9 


•974 


o- 100 


4 


•937 


5 


•037 





■980 


1 


020 





198 


O 


203 


2*4 


ii 


•023 


0-091 


5 


•466 


5 


•557 





■984 


1 


017 





180 


O 


183 


2-5 


12 


•182 


0-082 


6 


•050 


6 


•132 





■987 


1 


014 





163 


O 


165 


2*6 


!3 


.464 


0-074 


6 


•695 


6 


.769 





•989 


1 


on 





148 


O 


149 


2'7 


14 


•880 


0-067 


7 


•406 


7 


•473 





•991 


1 


■009 





■134 


O 


135 


2'8 


l6 


•445 


0-061 


8 


• 192 


8 


' 2 53 





■993 


1 


007 





121 


O 


122 


2*9 


18 


•174 


°'°55 


9 


•060 


9 


•115 





•994 


1 


006 





no 


O 


no 


3-o 


20 


•090 


0-050 


10 


•018 


10 


.068 





995 


1 


005 





°99 


O 


o99 


3-i 


22 


•198 


0-045 


11 


•076 


11 


• 122 





996 


1 


•004 





■090 


O 


•090 


3*2 


24 


•533 


0-041 


12 


•246 


12 


•287 





997 


1 


003 





081 


O 


082 


3*3 


27 


•113 


0-037 


13 


•538 


13 


•575 





•997 


1 


•003 





074 


O 


074 


3'4 


29 


■964 


0-033 


14 


•965 


14 


•999 





998 


1 


002 





067 


O 


067 


3*5 


33 


•116 


0-030 


16 


■543 


16 


573 





998 


1 


002 





060 


O 


060 


3-6 


36 


•598 


0-027 


18 


•286 


18 


3i3 





999 


1 


001 





o55 


O 


o55 


3' 7 


40 


■447 


0-025 


20 


•211 


20 


•236 





•999 


1 


001 





.049 


O 


049 


3*8 


44 


• 701 


0-022 


22 


'339 


22 


362 





999 


1 


001 





045 


O 


045 


3' 9 


49 


•402 


0-020 


24 


691 


24 


.711 





999 


1 


001 





040 


O 


041 


4-o 


54 


•598 


0-018 


27 


-290 


27 


•308 





999 


1 


001 





037 


O 


037 


\n 


4 


•810 


0-208 


2 


301 


2 


'5°9 





917 


1 


091 





398 


O 


435 


* 


23-141 


0*043 


n-549 


11-592 


0-996 


1-004 


0-087 


O 


■087 


If x: 


>4. 


then 


(approx 


ima 


;tely) 


sin 


h x = 


COS 


h x- 


-h 


?* = 


iti 


lie IS 


apj 


erian 



antilogarithm of x. 



TABLES. 



315 



5. LAMBDA FUNCTION. 



e 


M 

o-ooo 


e 

i5° 


KO) 
0-265 


6 
30° 


X 


(0) 



45° 


M 


e 

6o° 


KO) \ 



75° 


0° 


o-549 


o-88i 


i-3i7! 


1° 


0-017 


1 6° 


0-283 


3i° 





"57° 


46 


0-906 


6i° 


1 


'352 


76 


2° 


°'°3S 


! i7° 


0-301 


32 





59o 


47° 


°-93 2 , 


62 


1 


389 


77° 


3° 


0.052J 


! 18 


0-319 


33° 





611 


48 


o-957, 


63 


1 


4271 


78 


4° 


0-070 


l IQ° 


°'33% 


34° 





632 


49° 


0-984 


64 


1 


466! 


79° 


5° 


0-087 


20° 


'356 


35° 





6.S3 


50 


I -OIIi 


65 


1 


5o6j 


8o° 


6° 


0-105 


21° 


°'375 


36° 





674 


5i° 


1-038 


66° 


1 


549 


8i° 


7° 


O- 122 


22° 


o-394 


37° 





696 


52° 


i- 066 


67 


1 


592 


82 


8° 


o- 140 


23° 


0-413 


38 





718 


53° 


1-095 


68° 


1 


638 


83° 


9° 


0-I 5 8 


24° 


0-432 


39° 





740 


54° 


1 • 124 


69 


1 


686 


84 


10° 


O.I75 


25° 


0-451 


40 





763 


55° 


i-i54 


70 


I- 


735 


85 


n° 


0-IQ3 


26° 


0-470 


4i° 





786 


56 


i-i85 


7i° 


1 


788 


86° 


12° 


0-2II 


27 


0-490 


42 


o- 


809 


57° 


1 -217 


72 


i" 


843 


87 


13° 


0-229 


28 


0-509 


43° 


O' 


833 


58 


1-249 


73° 


i- 


901 


88° 


i 4 ° 


0-247 


29 


0-529 


44° 


o- 


857 


59° 


1-283 


74° 


I- 


962 


89 


15° 


0-265 


30 


o-549 


45° 


0.881 


6o° 


i-3i7 


75° 


2-028 


90 



X(d) 

2-028 
2-097 

2- 172 

2-253 
2-340 
2-436 
2-542 
2- 660 

2-794 
2.949 

3-I3I 

3*355 

3-643 

4-048 

4-741 

CO 



/(0) = log e tan(l;r+J0) = loge (sec 0+tan 0), X(-6)=-X(d), 0=gdX(6). 

6. GAMMA FUNCTION. 



n 



r(,) 



'OO 

•01 
.02 

•03 
•04 

•05 

-06 

.07 

.08. 
•09 
•10 

• n 

• 12 

•13 
•14 

•15 
•16 

•17 
•18 

•19 

•20 



•9943 
-9888 

•9836, 

•9784 

•9735 
.9688 

•9642 

•9597 
•9555 
•95 J 4 
•9474 
•9436 
•9399 
•9364 
•933o 
•9298 
•9267 

•9237 
•9209 

•9182 



n 



n ) 



I-20 


o- 


I-2I 


o- 


I -22 


o- 


1-23 


o- 


1-24 


o- 


1-25 


o- 


1-26 


o- 


1-27 


o- 


1.28 


o- 


1-29 


o- 


1-30 


o- 


i-3i 


o- 


1-32 


o- 


1 '33 


o- 


i-34 


o- 


i-35 


o- 


1-36 


o- 


i-37 


o- 


1-38 


o- 


i-39 


o- 


1 -40 


o- 



9182 

9156 

9 J 3i 
9108 

9085 

9064 

9044 
9025 
9007 
8990 [ 

8975| 
8960I 

8946, 

89341 
8922 

891 1 
8902 
8893 
8885 
8879 
8873 



n 



1-40 
1. 41 

1-42 

i-43 
1-44 

i-45 
1-46 

1.47 

1-48 

1-49 

1-50 

i-5i 

1-52 

J -53 

i-54 

i-55 
1-56 

i-57 
1-58 

i-59 
i- 60 



r( n ) 



n 



•8873 


i- 


•8868 


i- 


•8864 


1 • 


•8860 


1 • 


•8858 


1 • 


-88 S 6 


i- 


-88^6 


i- 


-88 S 6 


i- 


•8857 


i- 


•8860 


1 • 


•8862 


1 • 


■8866 


1 • 


•8870 


1 • 


•8876 


1 • 


•8882 


i- 


•8889 


i- 


-8896 


i- 


•8905 


i- 


•8914 


i- 


•8924 


i- 


•8935, 

1 


i- 



-6o 
•61 
•62 

•63 
•64 

•65 
•66 

.67 
•68 
.69 

•7o 
•7i 

•72 

•73 

•74 

•75 
.76 

•77 
•78 

•79 
•8o 



r( n ) 



0-8935 
0.8947 
0-8959 

0-8972 

0-8986 

0-9001 
0-9017 

0-9033 
0-9050 

0-9068 
0-9086 
0-9106 
0-9126 

0-9147 

0-9168 
0-9191 
0-9214 

0-9238 

0-9262 

0-9288 
0-9314 



n 



r( n ) 



80 
81 
82 

S3 
84 

86 

87i 
88 

89 
90 

9i 

92 
93 
94 
95 
96 
97 
98 

99 

00 



•93H 

•9341 

•9369 

•9397 
•9426 

•9456 
•9487 
.9518 

•955i 
•9584 
•9618 
•96^2 
•9688 

•9724 
.9761 

•9799 

•9837 
.9877 

.9917 
•9958 



fO0 



r(n) = 



x n ~ l e-x dx, r(n+l) = nr(n). 



316 



TABLES. 



7. FIRST ELLIPTIC INTEGRAL, F(m,6), w = sina. 



6 


a=o° 


a=i 5 ° 
o-ooo 


a=3o° 


«=45° 
o-ooo 


a=6o° 
o-ooo 


«=75° 


a=90° 


0° 


o-ooo 


o-ooo 


o-ooo 


o-ooo 


5° 


o 


•087 


0-087 


0-087 


0-087 


0-087 


0-087 





•087 


10° 


o 


•175 


0-175 


0-175 


0-175 


0-175 


0-175 





•175 


15° 


o 


•262 


0-262 


0-263 


o- 263 


0-264 


0-265 





•265 


20° 


o 


•349 


°'35° 


0-351 


o-353 


o-354 


0-356 





'356 


25° 


o 


•436 


°*437 


0-440 


o-443 


0-447 


0-450 





■451 


30° 


o 


5 2 4 


o-525 


0.529 


o-536 


0-542 


o-547 





'549 


35° 


o 


•611 


0-613 


0-620 


0-630 


0-641 


0-649 





•653 


40 


o 


698 


o- 702 


0-712 


0-727 


0-744 


o-757 





763 


45° 


o 


7*5 


0-790 


0-804 


0-826 


0-851 


0-873 





881 


5o° 


o 


•873 


0-879 


0-898 


0-928 


0-965 


0-997 


1 


on 


55° 


o 


960 


0-968 


o-993 


1-034 


1-085 


1 '^33 


1 


154 


6o° 


I 


047 


1-058 


1-090 


i- 142 


1-213 


1-284 


1 


3 T 7 


65° 


I- 


i34 


1 -147 


1-187 


1-254 


1*349 


i*453 


1- 


5°6 


70° 


I 


222 


1-237 


1-285 


1-370 


1-494 


1-647 


i- 


735 


75° 


I' 


309 


1-327 


I-385 


1-488 


1-649 


1-871 


2- 


028 


8o° 


I ■ 


396 


1-418 


1-485 


i- 608 


1-813 


2-134 


2- 


436 


85° 


I- 


484 


1-508 


I-585 


i-73i 


1-983 


2-437 


3- 


I3 1 


90 


i-57i 


1-598 


1-686 


1-854 


2*157 


2-768 


00 



8. SECOND ELLIPTIC INTEGRAL, E(m, d), m = sma. 



e 


a=o° 


o-ooo 


a =30° 
o-ooo 


«=45° 
o-ooo 


a=6o° 

o-ooo 


a= 75 ° 


a=go° 


0° 


o-ooo 


o-ooo 


o-ooo 


5° 


0-087 


0-087 


0-087 


0-087 


0-087 





.087 





■087 


10° 


0-175 


0-174 


0-174 


0-174 


0-174 





■ 174 





•174 


15° 


0-262 


o- 262 


o- 261 


o- 260 


o- 260 





•259 





■259 


20° 


o-349 


o-349 


o-347 


0-346 


o°344 





•342 





•342 


25° 


o-436 


o-435 


o-433 


0.430 


0-426 





424 





■423 


30° 


0-524 


0-522 


0-518 


0-512 


0-506 





• 502 





■500 


35° 


o-6n 


0-608 


0-602 


o-593 


0-583 





576 





574 


40° 


0-698 


0-695 


0.685 


0-672 


0-657 





647 





643 


45° 


0-785 


0-781 


0-767 


0-748 


0-728 





713 





707 


50° 


0-873 


o-866 


0.848 


0-823 


o-795 





774 





766 


55° 


0-960 


0-952 


0.928 


0-895 


0-859 


o- 


830 





819 


6o° 


1-047 


1-037 


i-oo8 


0-965 


0-918 


o- 


881 


o- 


866 


65° 


i-i34 


1 • 122 


i- 086 


1 '°33 


0-974 





926 


o- 


906 


70° 


1-222 


i- 206 


1-163 


1-099 


1 -027 


o- 


965 


o- 


940 


75° 


1-309 


1-291 


1 • 240 


1-163 


1-076 


o- 


999 


o- 


966 


8o° 


I-396 


i-375 


1. 316 


1-227 


i- 122 


I- 


028 


o- 


985 


85° 


I-484 


1*460 


1.392 


1-289 


1-167 


I" 


°53 


o- 


996 


90° 


I-57I 


1-544 


1.467 


J-35 1 


I-2II 


I- 


076 


I- 


000 



INDEX. 



(The references are to pages.) 



Acceleration, 73 
Amsler, planimeter, 301 

integrator, 302 
Anchor ring, 180, 193 
Area, 95, 134, 159, 190, 191, 299 

of any surface, 200 
Argument of function, 11 
Asymptotes, 57, 153 
Asymptotic circle, 155 

Barometric measurement 

of heights, 39 
Bernoulli's equation, 266 

Cardioid, 151, 160, 170 
Catenary, 209, 269, 308 
Cauchy, form of remainder, ZZb 
Centre of conic, 45 
of any curve, 290 
of curvature, 84, 88 
of gravity, 176, 304 
of quadric, 65 

Centroid, 177 

Circle, 139, 150 
of curvature, 85 

Circular functions, 28, 32, 311, 312 

Cissoid, 55, 61, 139 

Clairaut's equation, 268 

Companion to cycloid, 31, 140 

Compound interest problem, 269 

Concavity and convexity, 82 

Conchoid, 163 

Conical point, 64 

Conjugate point, 52 

Consecutive points, 85 

Constant of integration, 98 

Continuity, 12, 250 

Convergence, 211 

Gurvature, 82, 157, 207 

Curve of pursuit, 285 



Curve tracing 290 
Cusp, 52 

Cycloid, 29, 91, 140, 307 
companion to, 31, 140 
Cylinder, 145, 198, 199, 200 . 

Damped vibrations, 284 
Demoivre's theorem, 220 
Derivative, 13 
Differences, small, 47 
Differential, 15 

coefficient, 15 

of area, 38 

of exponentials, 25 

of hyperbolic functions, 34 

of logarithms, 25 

of power, product, quotient, 16 

partial, 41, 250 

successive, 68, 250 

total, 42, 253 
Differential equations, 259, 272 

exact, 263 

homogeneous, 262 

linear, 265, 275 

Element of integral, 94 

Ellipse, 70, 87, 88, 140, 151, 162, 
241,308 

Ellipsoid, 144 194. 195 

Elliptic integrals, 239, 316 

Envelopes, 171 

Epicycloid, 169, 20G 

Epitrochoid, 170 

Equations, differential, 259, 272 
solution of, by approximat on, 
49 

Euler, exponential formulae, 220 
theorem on homogeneous func- 
tions, 44, 255 

Evolute, 88, 175, 207 

317 



318 



INDEX. 



Folium, 53, 80, 163 
Fourier, series, 228 

theorem, 236 
Function, complementary, 278 

even, odd, 40 

implicit, 11 

single-valued , multiple-valued , 
12 

Gamma function, 132. 315 
Graphs, 11 

Guaermannian, 36, 298 
Guldin, properties of the centre 

of gravity, 178 
Gyration, radius of, 183 

Helix, 67, 148 

Huyghens, formula for circular 

arc, 222 
Hyperbola, 141, 151, 162 

rectangular, 23, 139, 150, 162 
Hyperbolic functions, 34, 294, 313 
Hypocycloid, 169 

four-cusped, 22, 46, 92, 136 

Indeterminate forms, 244 
Infinitesimals, 4 

equivalence of, 8 

orders of, 7 
Inflexion, point of, 82, 155 
Integral, definite, indefinite, 98 

double, 190 

triple, 194 
Integrals, definite, 129 

elliptic, 239 

fundamental, 99 

particular, 278 
Integration, 93 

approximate, 239 

by parts, 120 

by substitution, 114 

by successive reduction, 124 

of rational fractions, 112 

mechanical, 299 

successive, 189 
Integrator, 302 
Interval, 13 
Intrinsic equation, 205 
Inverse curves, 165 
Involute, 88 

of circle, 158, 207 

Kinetic energy of rotation, 181 



Lagrange, form of remainder, 225 
Lambda function, 105, 315 
Lemniscate, 53, 150, 156, 157, 161, 

242 
Lengths of curves, 134, 148, 241 
Limagon, 151, 162 
Limit, 1 

Lituus, 151, 154, 156 
Logarithms, calculation of, 216 
Napierian, 310, 311 

Maclaurin, series, 218, 226 

theorem, 226 
Maxima and minima, 75, 256 
Mean value, theorem of, 223 
Mean values, 202 
Moment, of area, 191, 302 

of inertia, 181, 191, 192, 197, 302 
Multiple points, 51, 56, 156 

Node, 52 

Normal, 21, 44, 150 
of surface, 64 

Pappus, properties of centre of 

gravity, 178 
Parabola, 22, 70, 90, 141, 151 

cubical, 136 

semicubical, 53, 136 
Paraboloid, elliptic, 144 

hyperbolic, 195 
Partial fractions, 287 
Pedal curves, 166 
Pendulum, 243 
Planimeter, 302 
Polar coordinates, 136 
Polar reciprocals, 168 
Potential, 193, 197 
Prismoidal formula, 147 
Product of inertia, 191 

Radius, of curvature, 85, 158 

of gyration, 183 
Rates, 72 

Rolle's theorem, 224 
Roulettes, 168 

Series, Fourier's, 228 
Gregory's, 217 
infinite, 211 
logarithmic, 216 
Maclaurin's, 218, 226 



INDEX. 



319 



Series, power, 213 
Taylor's, 220, 226 

Simpson's rule, 142 

Singular forms, 244 

Solid of revolution, 96, 134 

Sphere, 139, 196, 198 

Spheroids, 141 

Spiral, of Archimedes, 151, 160 
hyperbolic, 151, 154 
logarithmic, 152, 153. 158 

Subnormal, 21, 150 

Subtangent, 21, 150 

Symmetry, 290 

Tangent, 20, 44, 150 

to curve in space, 65 
Tangent plane, 62 



Taylor, series, 220 226 

theorem, 223 
Torus, 180, 193 
Tractrix, 207 
Trajectories, 270 
Triple point, 52 
Trochoid, 168 
Turning value, 76 

Value of it, 217 

Variable, 1, 11 
change of, 280 
independent, 11, 68 

Velocity, 73 

Volumes, 134, 144, 193, 196 

Witch.. 84, 139 



SHORT-TITLE CATALOGUE 

OP THE 

PUBLICATIONS 

OP 

JOHN WILEY & SONS, 

New York. 
London: CHAPMAN & HALL, Limited. 



ARRANGED UNDER SUBJECTS. 



Descriptive circulars sent on application. Books marked with an asterisk (*) are sold 
at «^/ ? prices only, a double asterisk (**) books sold under the rules of the American 
Publishers' Association at net prices subject to an extra charge for postage. All books 
are bound in cloth unless otherwise stated. 



AGRICULTURE. 

Armsby's Manual of Cattle-feeding i2mo, $i 75 

Principles of Animal Nutrition 8vo, 4 00 

Budd and Hansen's American Horticultural Manual: 

Part I. Propagation, Culture, and Improvement i2mo, 1 50 

Part II. Systematic Pomology i2mo, 1 50 

Downing's Fruits and Fruit-trees of America 8vo, 5 00 

Elliott's Engineering for Land Drainage i2mo, 1 50 

Practical Farm Drainage i2mo, 1 00 

Graves's Forest Mensuration 8vo, 4 00 

Green's Principles of American Forestry i2mo, 1 50 

Grotenfelt's Principles of Modern Dairy Practice. (Woll.) i2mo, 2 00 

Kemp's Landscape Gardening i2mo, 2 50 

Maynard's Landscape Gardening as Applied to Home Decoration i2mo, 1 50 

* McKay and Larsen's Principles and Practice of Butter-making 8vo : 1 50 

Sanderson's Insects Injurious to Staple Crops i2mo, 1 50 

Insects Injurious to Garden Crops. (In preparation.) 
Insects Injuring Fruits. (In preparation.) 

Stockbridge's Rocks and Soils 8vo, 2 50 

Winton's Microscopy of Vegetable Foods 8vo, 7 50 

Woll's Handbook for Farmers and Dairymen i6mo, 1 50 



ARCHITECTURE. 

Baldwin's Steam Heating for Buildings i2mo, 2 50 

Bashore's Sanitation of a Country House nmo, 1 00 

Berg's Buildings and Structures of American "Railroads 4to, 5 00 

Birkmire's Planning and Construction of American Theatres 8vo, 3 00* 

Architectural Iron and Steel 8vo, 3 50* 

Compound Riveted Girders as Applied in Buildings 8vo, 2 00 

. Planning and Construction of High Office Buildings 8vo, 3 50 

Skeleton Construction in Buildings 8vo, 3 00 

Brigg's Modern American School Buildings 8vo, 4 00 

\ 



Carpenter's Heating and Ventilating of Buildings 8vo, 4 00 

Freitag's Architectural Engineering 8vo, 3 50 

Fireproofing of Steel Buildings 8vo, 2 50 

French and Ives's Stereotomy 8vo, 2 50 

Gerhard's Guide to Sanitary House-inspection i6mo, 1 00 

Theatre Fires and Panics i2mo, 1 50 

*Greene's Structural Mechanics 8vo, 2 50 

Holly's Carpenters' and Joiners' Handbook i8mo, 75 

Johnson's Statics by Algebraic and Graphic Methods 8vo, 2 00 

Kidder's Architects' and Builders' Pocket-book. Rewritten Edition. i6mo,mor., 5 00 

Merrill's Stones for Building and Decoration 8vo, 5 00 

Non-metallic Minerals: Their Occurrence and Uses 8vo, 4 00 

Monckton's Stair-building 4to, 4 00 

Patton's Practical Treatise on Foundations 8vo, 5 00 

Peabody's Naval Architecture 8vo, 7 50 

Rice's Concrete -block Manufacture 8vo, 2 00 

Richey's Handbook for Superintendents of Construction i6mo, mor., 4 00 

* Building Mechanics' Ready Reference Book. Carpenters' and Wood- 
workers' Edition i6mo, morocco, 1 50 

Sabin's Industrial and Artistic Technology of Paints and Varnish 8vo, 3 00 

Siebert and Biggin's Modern Stone-cutting and Masonry 8vo, 1 50 

Snow's Principal Species of Wood 8vo, 3 50 

Sondericker's Graphic Statics with Applications to Trusses, Beams, and Arches. 

8vo, 2 00 

Towne's Locks and Builders' Hardware i8mo, morocco, 3 00 

Wait's Engineering and Architectural Jurisprudence 8vo, 6 00 

Sheep, 6 50 

Law of Operations Preliminary to Construction in Engineering and Archi- 
tecture 8vo, 5 00 

Sheep, 5 50 

Law of Contracts 8vo, 3 00 

Wood's Rustless Coatings: Corrosion and Electrolysis of Iron and Steel. .8vo, 4 00 
Woicester and Atkinson's Small Hospitals, Establishment and Maintenance, 
Suggestions for Hospital Architecture, with Plans for a Small Hospital. 

i2mo, 1 25 

The World's Columbian Exposition of 1893 Large 4to, 1 00 



ARMY AND NAVY. 

Bernadou's Smokeless Powder, Nitro-cellulose, and the Theory of the Cellulose 

Molecule i2mo, 2 50 

* Bruff 's Text-book Ordnance and Gunnery 8vo, 6 00 

Chase's Screw Propellers and Marine Propulsion 8vo, 3 00 

Cloke's Gunner's Examiner 8vo, 1 50 

Craig's Azimuth. 4to, 3 50 

Crehore and Squier's Polarizing Photo-chronograph 8vo, 3 00 

* Davis's Elements of Law 8vo, 2 50 

* Treatise on the Military Law of United States 8vo, 7 00 

Sheep, 7 50 

De Brack's Cavalry Outposts Duties. (Carr.) 24010, morocco, 2 00 

Dietz's Soldier's First Aid Handbook i6mo, morocco, 1 25 

* Dredge's Modern French Artillery 4to, half morocco, 15 00 

Durand's Resistance and Propulsion of Ships 8vo, 5 00 

* Dyer's Handbook of Light Artillery i2mo, 3 00 

Eissler's Modern High Explosives 8vo, 4 00 

* Fiebeger's Text-book on Field Fortification .Small 8vo, 2 00 

Hamilton's The Gunner's Catechism i8mo, 1 1 

* Soft's Elementary Naval Tactics. . . .8vo, 1 §o 



I 


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Ingalls's Handbook of Problems in Direct Fire 8vo, 4 00 

* Ballistic Tables 8vo, 

* Lyons's Treatise on Electromagnetic Phenomena. Vols. I. and II. .8vo, each, 

* Mahan's Permanent Fortifications. (Mercur.) 8vo, half morocco, 

Manual for Courts-martial i6mo, morocco, 

* Mercur's Attack of Fortified Places 121110, 

* Elements of the Art of War . . 8vo. 

Metcalf's Cost of Manufactures — And the Administration of Workshops. .8vo, 

* Ordnance and Gunnery. 2 vols i2mo, 

Murray's Infantry Drill Regulations i8mo, paper, 

Nixon's Adjutants' Manual 241110, 

Peabody's Naval Architecture 8vo, 

* Phelps's Practical Marine Surveying • 8vo, 

Powell's Army Officer's Examiner i2mo, 

Sharpe's Art of Subsisting Armies in War i8mo, morocco, 

* Tupes and Poole's Manual of Bayonet Exercises and Musketry Fencing. 

24mo, leather, 

* Walke's Lectures on Explosives 8vo, 

Weaver's Military Explosives 8vo, 

* Wheeler's Siege Operations and Military Mining 8vo, 

Winthrop's Abridgment of Military Law • ••«•, i2mo, 

Woodhull's Notes on Military Hygiene i6mo, 

Yovng'«; Simple Elements of Navigation i6mo, morocco, 



ASSAYING. 

Fletcher's Practical Instructions in Quantitative Assaying with the Blowpipe. 

i2mo, morocco, 

Furman's Manual of Practical Assaying 8vo, 

Lodge's Notes on Assaying and Metallurgical Laboratory Experiments. . . .8vo, 

Low's Technical Methods of Ore Analysis 8vo, 

Miller's Manual of Assaying ..- i2mo, 

Cyanide Process i2rco, 

Minet's Production of Aluminum and its Industrial Use. (Waldo.) i2mo, 

O'Driscoll's Notes on the Treatment of Gold Ores 8vo, 

Ricketts and Miller's Notes on Assaying 8vo ; 

Robine and Lenglen's Cyanide Industry. (Le Clerc.) 8vo, 

Ulke's Modern Electrolytic Copper Refining. ....." 8vo, 

Wilson's Cyanide Processes. i2mo, 

Chlorination Process i2mo, 



ASTRONOMY. 

Comstock's Field Astronomy for Engineers 8vo, 2 50 

Craig's Azimuth 4to, 3 50 

Doolittle's Treatise on Practical Astronomy 8vo, 4 00 

Gore's Elements of Geodesy. 8vo, 2 50 

Hayford's Text-book of Geodetic Astronomy 8vo, 3 00 

Merriman's Elements of Precise Surveying and Geodesy 8vo, 2 50 

* Michie and Harlow's Practical Astronomy .8vo, 3 00 

* White's Elements of Theoretical and Descriptive Astronomy 121110, 2 00 



BOTANY. 

Davenport's Statistical Methods, with Special Reference to Biological Variation. 

i6mo, morocco, 1 25 

Thome and Bennett's Structural and Physiological Botany, i6mo, 2 25 

Westermaier's Compendium of General Botany. (Schneider.). > 8vo, 2 00 

3 



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CHEMISTRY. 

Adriance's Laboratory Calculations and Specific Gravity Tables i2mo, i 25 

Alexeyeff's GenerahPrinciples of Organic Synthesis. (Matthews.) 8\o, 3 00 

Allen's Tables for Iron Analysis 8vo, 3 00 

Arnold's Compendium of Chemistry. (Mandel.) Small 8vo. 3 50 

Austen's Notes for Chemical Students i2mo, 1 5a 

Bernadou's Smokeless Powder.— Nitro-cellulose, and Theory of the Cellulose 

Molecule i2mo, 2 50 

* Browning's Introduction to the Rarer Elements 8vo, 1 50 

Brush and Penfield's Manual of Determinative Mineralogy 8vo, 4 00 

Claassen's Beet-sugar Manufacture. (Hall and Rolfe.) . 8vo, 3 00 

Classen's Quantitative Chemical Analysis by Electrolysis. (Eoltwcod.). .8vo, 3 00 

Cohn's Indicators and Test-papers i2mo, 2 00 

Tests and Reagents 8vo, 3 00 

Crafts's Short Course in Qualitative Chemical Analysis. (Schaeffer.). . .i2mo, 1 50 
Dolezalek's Theory of the Lead Accumulator (Storage Battery). (Von 

Ende») i2mo, 2 50 

Drechsei's Chemical Reactions. (Merrill.) i2mo, 1 25 

Duhem's Thermodynamics and Chemistry. (Burgess.) 8vo, 4 00 

Eissler's Modern High Explosives .8vo, 4 00 

Effront's Enzymes and their Applications. (Prescott.) 8vOj 3 oc» 

Erdmann's Introduction to Chemical Preparations. (Dunlap.). i2mo, 1 25 

Fletcher's Practical Instructions in Quantitative Assaying with the Blowpipe. 

i2mo, morocco, 1 50 

Fowler's Sewage Works Analyses i2mo, 2 oo- 

Fresenius's Manual of Qualitative Chemical Analysis. (Wells.) 8vo, 5 00 

Manual of Qualitative Chemical Analysis. Part I. Descriptive. (Wells.) 8vo, 3 oo> 

System of Instruction in Quantitative Chemical Analysis. (Cohn.) 

2 vols 8vo, 12 50 

Fuertes's Water and Public Health i2mo, 1 50 

Furman's Manual of Practical Assaying 8vo, 3 00 

* Getman's Exercises in Physical Chemistry. . i2mo, 2 00 

Gill's Gas and Fuel Analysis for Engineers i2mo, 1 25 

Grotenfelt's Principles of Modern Dairy Practice. (Woll.) nmo, 2 00 

Groth's Introduction to Chemical Crystallography (Marshall) i2mo, 1 25 

Hammarsten's Text-book of Physiological Chemistry. (Mandel.) 8vo, 4 00 

Helm's Principles of Mathematical Chemistry. (Morgan.) i2mo, 1 50 

Hering's Ready Reference Tables (Conversion Factors) i6mo, morocco, 2 50 

Hind's Inorganic Chemistry 8vo, 3 o~ 

* Laboratory Manual for Students i2mo, 1 00 

Holleman's Text-book of Inorganic Chemistry. (Cooper.) 8vo, 2 50 

Text-book of Organic Chemistry. (Walker and Mott.) 8vo, 2 50 

* Laboratory Manual of Organic Chemistry. (Walker.). . v i2mo, 1 00 

Hopkins's Oil-chemists' Handbook 8vo, 3 00 

Jackson's Directions for Laboratory Work in Physiological Chemistry. .8vo, 1 25 

Keep's Cast Iron 8vo, 2 50 

Ladd's Manual of Quantitative Chemical Analysis i2mo, 1 co 

Landauer's Spectrum Analysis. (Tingle.) 8vo, 3 00 

* Langworthy and Austen. The Occurrence of Aluminium in Vegetable 

Products, Animal Products, and Natural Waters 8vo, 2 oo 

Lassar-Cohn's Practical Urinary Analysis. (Lorenz.) i2mo, 1 oo- 

Application of Some General Reactions to Investigations in Organic 

Chemistry. (Tingle.) i2mo, 1 00 

Leach's The Inspection and Analysis of Food with Special Reference to State 

Control 8vo, 7 50 

Lob's Electrochemistry of Organic Compounds. (Lorenz.) 8vo, 3 00 

Lodge's Notes on Assaying and Metallurgical Laboratory Experiments. .. .8vo, 3 00 

Low's Technical Method of Ore Analysis 8vo. 3 00 

Lunge's Techno-chemical Analysis. (Cohn.) i2mo 1 00 

4 



* McKay and Larsen's Principles and Practice of Butter-making 8vo 

Mandel's Handbook for Bio-chemical Laboratory i2mo 

* Martin's Laboratory Guide to Qualitative Analysis with the Blowpipe. . i2mo 
Mason's Water-supply. (Considered Principally from a Sanitary Standpoint.) 

3d Edition, Rewritten 8vo 

Examination of Water. (Chemical and Bacteriological.). . . i2mo 

Matthew's The Textile Fibres : . . .8vo 

Meyer's Determination of Radicles in Carbon Compounds. (Tingle.). .i2mo 

Miller's Manual of Assaying i2mo 

Cyanide Process . i2mo 

Minet's Production of Aluminum and its Industrial Use. (Waldo.) . . . . i2mo 

Mixter's Elementary Text-book of Chemistry i2mo 

Morgan's An Outline of the Theory of Solutions and its Results i2mo 

Elements of Physical Chemistry i2mo 

* Physical Chemistry for Electrical Engineers i2mo 

Morse's Calculations used in Cane-sugar Factories i6mo, morocco 

Muliiken's General Method for the Identification of Pure Organic Compounds 

Vol. I. . Large 8vo 

O'Brine's Laboratory Guide in Chemical Analysis 8vo 

O'Driscoll's Notes on the Treatment of Gold Ores 8vo 

Ostwald's Conversations on Chemistry. Part One. (Ramsey.). ..... . 12.U10 

" " Part Two. (Turnbull.) i2mo 

* Penfield's Notes on Determinative Mineralogy and Record of Mineral Tests 

8vo, paper 

Pictet's The Alkaloids and their Chemical Constitution. (Biddle.) 8vo 

Pinner's Introduction to Organic Chemistry. (Austen.) i2mo 

Poole's Calorific Power of Fuels 8vo 

Prescott and Winslow's Elements of Water Bacteriology, with Special Refer- 
ence to Sanitary Water Analysis i2mo 

* Reisig's Guide to Piece-dyeing 8vo 

Richards and Woodman's Air.Water, and Food from a Sanitary Standpoint. ,8vo 
Ricketts and Russell's Skeleton Notes upon Inorganic Chemistrjr. (Part ] 

Non-metallic Elements.) 8vo, morocco 

Ricketts and Miller's Notes on Assaying 8vo 

Rideal's Sewage and the Bacterial Purification of Sewage 8vo 

Disinfection and the Preservation of Food 8vo 

Riggs's Elementary Manual for the Chemical Laboratory 8vo 

Robine and Lenglen's Cyanide Industry. (Le Clerc.) '. 8vo 

Rostoski's Serum Diagnosis. (Bolduan.) i2mo 

Ruddiman's Incompatibilities in Prescriptions 8vo 

* Whys in Pharmacy , i2mo 

Sabin's Industrial and Artistic Technology of Paints and Varnish 8vo 

Salkowski's Physiological and Pathological Chemistry. (Orndorff.) 8vo 

Schimpf's Text-book of Volumetric Analysis. * '. i2mo 

Essentials of Volumetric Analysis i2mo 

* Qualitative Chemical Analysis 8vo 

Smith's Lecture Notes on Chemistry for Dental Students .8vo 

Spencer's Handbook for Chemists of Beet-sugar Houses i6mo, morocco 

Handbook for Cane Sugar Manufacturers i6mo, morocco 

Stockbridge's Rocks and Soils 8vo 

* Tillman's Elementary Lessons in Heat 8vo 

* Descriptive General Chemistry 8vo 

Treadwell's Qualitative Analysis. (Hall.) 8vo 

Quantitative Analysis. (Hall.) 8vo 

Turneaure and Russell's Public Water-supplies 8vo 

Van Deventer's Physical Chemistry for Beginners. (Boltwood.) i2mo 

* Walke's Lectures on Explosives 8vo 

Ware's Beet-sugar Manufacture and Refining . .Small 8vo, cloth 

Washington's Manual of the Chemical Analysis of Rocks 8vo 

5 



1 50 

1 50 

60 

4 00 
1 25 

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3 00 
I 50 

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5 00 

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1 50 

2 00 

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1 25 

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Wassermann's Immune Sera : Hemolysins, Cytotoxins, and Precipitins. (Bol- 

duan -> *.i2mo, 

"Weaver's Military Explosives g vo 

Wehrenfennig's Analysis and Softening of Boiler Feed- Water 8vo. 

Wells's Laboratory Guide in Qualitative Chemical Analysis 8vo. 

Short Course in Inorganic Qualitative Chemical Analysis for Engineering 
Students i2mo, 

Text-book of Chemical Arithmetic i2mo, 

Whipple's Microscopy of Drinking-water 8vo, 

Wilson's Cyanide Processes _ i2mo 

Chlorination Process i2mo 

Winton's Microscopy of Vegetable Foods • 8vo, 

Wulling's Elementary Course in Inorganic, Pharmaceutical, and Medical 

Chemistry, i 2 mo, 2 00 



CIVIL ENGINEERING. 

BRIDGES AND ROOFS. HYDRAULICS. MATERIALS OF ENGINEERING. 

kAILWAY ENGINEERING. 

Baker's Engineers' Surveying Instruments i2mo, 

Bixby's Graphical Computing Table Paper 19^X24! inches. 

^* Burr's Ancient and Modem Engineering and the Isthmian Cana .. (Postage, 

27 cents additional.). 8vo, 

Comstock's Field Astronomy for Engineers 8vo, 

-Davis's Elevation and Stadia Tables 8vo, 

Elliott's Engineering for Land Drainage i2mo, 

Practical Farm Drainage i2mo, 

*Fiebeger's Treatise on Civil Engineering 8vo, 

Flemer's Phototopographic Methods and Instruments 8vo, 

Folwell's Sewerage. (Designing and Maintenance.) 8vo, 

Freitag's Architectural Engineering. 2d Edition, Rewritten 8vo, 

French and Ives's Stereotomy 8vo, 

Ooodhue's Municipal Improvements i2mo, 

Goodrich's Economic Disposal of Towns' Refuse 8vo, 

Gore's Elements of Geodesy 8vo, 

Hayford's Text-book of Geodetic Astronomy 8vo, 

Hering's Ready Reference Tables (Conversion Factors) i6mo, morocco, 

-Howe's Retaining Walls for Earth i2rno, 

^ Ives's Adjustments of the Engineer's Transit and Level i6mo, Bds. 

Ives and Hilts's Problems in Surveying i6mo, morocco, 

Johnson's (J. B.) Theory and Practice of Surveying Small 8vo, 

Johnson's (L. J.) Statics by Algebraic and Graphic Methods 8vo, 

Laplace's Philosophical Essay on Probabilities. (Truscott and Emory.) . i2mo, 

Mahan's Treatise on Civil Engineering. (1873.) (Wood.) 8vo, 

* Descriptive Geometry 8vo, 

Merriman's Elements of Precise Surveying and Geodesy 8vo, 

Merriman and Brooks's Handbook for Surveyors. i6mo, morocco, 

Tiugent's Plane Surveying 8vo, 

Ogden's Sewer Design , i2mo, 

Parsons's Disposal of Municipal Refuse 8vo, 

Patton's Treatise on Civil Engineering 8vo half leather, 

Reed's Topographical Drawing and Sketching 4 to, 

Hideal's Sewage and the Bacterial Purification of Sewage 8vo, 

Siebert and Biggin's Modern Stone-cutting and Masonry 8vo, 

Smith's Manual of Topographical Drawing. (McMillan.) 8vc, 

Sondericker's Graphic Statics, with Applications to Trusses, Beams, and Arches. 

8vo, 2 00 

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Taylor and Thompson's Treatise on Concrete, Plain and Reinforced 8vo, 

* Trautwine's Civil Engineer's Pocket-book i6mo, morocco, 

Venable's Garbage Crematories in America 8vo, 

Wait's Engineering and Architectural Jurisprudence 8vo, 

Sheep, 
Law of Operations Preliminary to Construction in Engineering and Archi- 
tecture , 8vo, 

Sheep, 

Law of Contracts 8vo, 

Warren's Stereotomy — Problems in Stone-cutting 8vo, 

Webb's Problems in the Use and Adjustment of Engineering Instruments. 

i6mo, morocco, 

Wilson's Topographic Surveying 8vo, 



BRIDGES AND ROOFS. 

Boiler's Practical Treatise on the Construction of Iron Highway Bridges. .8vo, 2 00 

+ Thames River Bridge 4to, paper, 5 00 

Burr's Course on the Stresses in Bridges and Roof Trusses, Arched Ribs, and 

Suspension Bridges 8vo, 3 50 

Burr and Falk's Influence Lines for Bridge and Roof Computations 8vo, 3 oo 

Design and Construction of Metallic Bridges 8vo, 5 oa 

Du Bois's Mechanics of Engineering. Vol. II £rr.all 4to, 10 co 

Foster's Treatise on Wooden Trestle Bridges 4to, 5 00 

Fowler's Ordinary Foundations 8vo, 3 50 

Greene's Roof Trusses 8vo, 1 2s 

Bridge Trusses 8vo, 2 50* 

Arches in Wood, Iron, and Stone 8vo s 2 50= 

Howe's Treatise on Arches 8vo, 4 oa 

Design of Simple Roof- trusses in Wood and Steel 8vo, 2 oa 

Symmetrical Masonry Arches 8vo, 2 50 

Johnson, Bryan, and Turneaure's Theory and Practice in the Designing of 

Modern Framed Structures Small 4to, 10 oa 

Merriman and Jacoby's Text-book on Roofs and Bridges : 

Part I. Stresses in Simple Trusses 8vo, 2 sot- 
Part II. Graphic Statics 8vo, 2 5a 

Part III. Bridge Design. 8vo, 2 50- 

Part IV. Higher Structures 8vo, 2 50 

Morison's Memphis Bridge 4*0, 10 00 

Waddell's De Pontibus, a Pocket-book for Bridge Engineers. . i6iro, morocco, 2 00 

* Specifications for Steel Bridges i2mo, 50 

Wright's Designing of Draw-spans. Two parts in one volume 8vo, 3 5a 



HYDRAULICS. 

Barnes's Ice Formation 8vo, 3 oa 

Bazin's Experiments upon the Contraction of the Liquid Vein Issuing from 

an Orifice. (Trautwine.) 8vo, 

Bovey's Treatise on Hydraulics 8vo, 

Church's Mechanics of Engineering 8vo, 

Diagrams of Mean Velocity of Water in Open Channels paper, 

Hydraulic Motors 8vo, 

Coffin's Graphical Solution of Hydraulic Problems i6mo, morocco, 

Flather's Dynamometers, and the Measurement of Power i2mo, 

Folwell's Water-supply Engineering 8vo, 

Frizell's Water-power 8vo, 

7 



2 


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I^uertes's Water and Public Health , • i2mo, 1 50 

Water-filtration Works. i2mo, 2 50 

Canguillet and Kutter's General Formula for the Uniform Flow of Water in 

Rivers and Other Channels. (Hering and Trautwine.) 8vo, 4 00 

Hazen's Filtration of Public Water-supply 8vo, 3 00 

Eazlehurst's Towers and Tanks for Water-works 8vo, 2 50 

Herschel's 115 Experiments on the Carrying Capacity of Large, Riveted, Metal 

Conduits 8vo ; 2 00 

Mason's Water-supply. (Considered Principally from a Sanitary Standpoint.) 

8vo, 

Merriman's Treatise on Hydraulics 8vo, 

"* Michie's Elements of Analytical Mechanics . 8vo, 

Schuyler's Reservoirs for Irrigation, Water-power, and Domestic Water- 
supply Large 8vo, 

** Thomas and Watt's Improvement of Rivers (Post., 44c. additional.) 4to, 

Turneaure and Russell's Public Water-supplies 8vo, 

Wegmann's Design and Construction of Dams 4to, 

Water-supply of the City of New York from 1658 to 1895. ^to, 

Williams and Hazen's Hydraulic Tables 8vo, 

Wilson's Irrigation Engineering Small 8vo, 

Wolff's Windmill as a Prime Mover 8vo, 

Wood's Turbines 8vo, 

Elements of Analytical Mechanics 8vo, 



MATERIALS OF ENGINEERING. 

Baker's Treatise on Masonry Construction 8vo, 

Roads and Pavements 8vo, 

Black's United States Public Works Oblong 4to, 

-* Bovey's Strength of Materials and Theory of Structures 8vo, 

Burr's Elasticity and Resistance of the Materials of Engineering 8vo, 

Byrne's Highway Construction 8vo, 

Inspection of the Materials and Workmanship Employed in Construction. 

i6mo, 

Church's Mechanics of Engineering 8vo, 

Du Bois's Mechanics of Engineering. Vol. I Small 4to, 

^Eckel's Cements, Limes, and Plasters 8vo, 

Johnson's Materials of Construction Large 8vo, 

Towler's Ordinary Foundations 8vo, 

Graves's Forest Mensuration 8vo, 

* Greene's Structural Mechanics 8vo, 

Xeep's Cast Iron 8vo, 

Lanza's Applied Mechanics 8vo, 

Marten's Handbook on Testing Materials. (Henning.) 2 vols 8vo, 

Maurer's Technical Mechanics 8vo, 

Merrill's Stones for Building and Decoration 8vo, 

Merriman's Mechanics of Materials .... 8vo, 

Strength of Materials ' i2mo, 

Metcalf's Steel. A Manual for Steel-users i2mo, 

Patton's Practical Treatise on Foundations 8vo, 

Richardson's Modern Asphalt Pavements „ 8vo, 

Richey's Handbook for Superintendents of Construction i6mo, mor., 

* Ries's Clays: Their Occurrence, Properties, and Uses. . 8vo, 

Rockwell's Roads and Pavements in France i2mo, 

Sabin's Industrial and Artistic Technology of Paints and Varnish 8vo, 

Smith's Materials of Machines i2mo f 

.Snow's Principal Species of Wood 8vo, 

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Spalding's Hycbaulic Cement i2mo, 

Text-book on Roads and Pavements ... i2mo, 

Taylor and Thompson's Treatise on Concrete. Plain and Reinforced 8vo, 

Thurston's Materials of Engineering. 3 Parts 8vo, 

Part I. Non-metallic Materials of Engineering and Metallurgy 8vo, 

Part II Iron and Steel 8vo, 

Part III. A Treatise on Brasses, Bronzes, and Other Alloys and their 

Constituents. 8vo, 

Thurston's Text-book of the Materials of Construction 8vo, 

Tillson's Street Pavements and Paving Materials 8vo, 

Waddell's De Pontibus (A Pocket-book for Bridge Engineers.) . . i6mo, mor., 

Specifications for Steel Bridges i2mo, 

Wood's (De V.) Treatise on the Resistance of Materials, and an Appendix on 

the Preservation of Timber 8vo, 

vVood's (De V.) Elements of Analytical Mechanics 8vo, 

"Wood's (M. P.) Rustless Coatings: Corrosion and Electrolysis of Iron and 

Steel 8vo, 4 00 



RAILWAY ENGINEERING. 

Andrew's Handbook for Street Railway Engineers,. . . .3x5 inches, morocco, 

Berg's. Buildings and Structures of American Railroads 4to, 

Brook's Handbook of Street Railroad Location i6mo, morocco , 

Butt's Civil Engineer's Field-book i6mo, morocco. 

Crandall's Transition Curve i6mo, morocco, 

Railway and Other Earthwork Tables 8vo. 

Dawson's "Engineering" and Electric Traction Pocket-book i6mo r morocco, 
Dredge's History of the Pennsylvania Railroad: (1879) Paper, 

* Drinker's Tunnelling, Explosive Compounds, and Rock Drills. 4to, half mor., 

Fisher's Table of Cubic Yards Cardboard, 

Godwin's Railroad Engineers' Field-book and Explorers' Guide. . . i6mo, mor., 

Howard's Transition Curve Field-book i6mo, morocco, 

Hudson's Tables for Calculating the Cubic Contents of Excavations and Em- 
bankments. 8vo, 

Moli tor and Beard's Manual for Resident Engineers. = i6mo, 

Nagle's Field Manual for Railroad Engineers i6mo, morocco* 

Philbrick's Field Manual for Engineers. . i6mo, morocco, 

Searles's Field Engineering i6mo, morocco, 

Railroad Spiral i6mo, morocco, 

Taylor's Prismoidal Formulas and Earthwork. . , 8vo, 

* Trautwine's Method of Calculating the Cube Contents of Excavations and 

Embankments by the Aid of Diagrams 8vo, 2 00 

The Field Practice of Laying Out Circular Curves for Railroads. 

i2mo, morocco, 2 5c 

Cross-section Sheet Paper, 25 

Webb's Railroad Construction i6mo, morocco, 5 00 

Economics of Railroad Construction Large i2mo, 2 50 

Wellington's Economic Theory of the Location of Railways. . . . .Small 8vo- 5 00 



DRAWING. 

Barr's Kinematics of Machinery 8vo 2 50 

* Bartlett's Mechanical Drawing 8vo, 3 00 

* " <4 " Abridged Ed 8vo, 150 

Coolidge's Manual of Drawing 8vo, paper, 1 00 



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Coolidge and Freeman's Elements of General Drafting for Mechanical Engi- 
neers Oblong 4to 

Durley's Kinematics of Machines 8vo 

Emch's Introduction to Projective Geometry and its Applications 8vo 

Hill's Text-book on Shades and Shadows, and Perspective 8vo 

Jamison's Elements of Mechanical Drawing 8vo 

Advanced Mechanical Drawing 8vo 

Jones's Machine Design: 

Part I. Kinematics of Machinery 8vo 

Part II. Form, Strength, and Proportions of Parts 8vo 

MacCord's Elements of Descriptive Geometry 8vo 

Kinematics; or, Practical Mechanism 8vo 

Mechanical Drawing 4to 

Velocity Diagrams 8vo 

MacLeod's Descriptive Geometry Small 8vo 

* Mahan's Descriptive Geometry and Stone-cutting 8vo 

Industrial Drawing. (Thompson.) 8vo 

Moyer's Descriptive Geometry 8vo 

Reed's Topographical Drawing and Sketching 4to 

Reid's Course in Mechanical Drawing 8vo 

Text-book of Mechanical Drawing and Elementary Machine Design. 8vo 

Robinson's Principles of Mechanism 8vo 

Schwamb and Merrill's Elements of Mechanism 8vo 

Smith's (R. S.) Manual of Topographical Drawing. (McMillan.). 8vo 

Smith (A. W.) and Marx's i T achine Design 8vo 

* Titsworth's Elements of Mechanical Drawing Oblorg 8vo 

Warren's Elements of Plane and Solid Free-hand Geometrical Drawing. i2mo 

Drafting Instruments and Operations i2mo 

Manual of Elementary Projection Drawing i2mo 

Manual of Elementary Problems in the Linear Perspective of Form and 

Shadow i2mo 

Plane Problems in Elementary Geometry. . . . i2mo 

Primary Geometry i2mo 

Elements of Descriptive Geometry, Shadows, and Perspective 8vo 

General Problems of Shades and Shadows . . , 8vo 

Elements of Machine Construction and Drawing . . 8vo 

Problems, Theorems, and Examples in Descriptive Geometry 8vo 

Weisbach's Kinematics and Power of Transmission. (Hermann and 

Klein.). . 8vo 

Whelpley's Practical Instruction in the Art of Letter Engraving i2mo 

Wilson's (H. M.) Topographic Surveying 8vo 

Wilson's (V. T.) Free-hand Perspective 8vo 

Wilson's (V, T.) Free-hand Lettering. . , 8vo 

Woolf 's Elementary Course in Descriptive Geometry Large 8vo 



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ELECTRICITY AND PHYSICS. 



Anthony and Brackett's Text-book of Physics. (Magie.) Small 8vo, 

Anthony's Lecture-notes on the Theory of Electrical Measurements. . . . i2mo, 

Benjamin's History of Electricity 8vo, 

Voltaic Cell 8vo, 

Classen's Quantitative Chemical Analysis by Electrolysis. (Boltwood.).8vo, 

* Collins's Manual of Wireless Telegraphy i2mo, 

Morocco, 

Crehore and Squier's Polarizing Photo-chronograph. . , 8vo, 

Dawson's "Engineering" and Electric Traction Pocket-book. i6mo, morocco, 

10 



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5 


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Dolezalek's Theory of the Lead Accumulator (Storage Battery). (Von 

Ende.) i2mo, 

Duhem's Thermodynamics and Chemistry. (Burgess.) 8vo, 

Flather's Dynamometers, and the Measurement of Power i2mo, 

Gilbert's De Magnete. (Mottelay.) 8vo, 

Hanchett's Alternating Currents Explained i2mo, 

Hering's Ready Reference Tables (Conversion Factors) i6mo, morocco, 

Holman's Precision of Measurements 8vo, 

Telescopic Mirror-scale Method, Adjustments, and Tests. . . .Large 8vo, 

Kinzbrunner's Testing of Continuous-current Machines 8vo, 

Landauer's Spectrum Analysis. (Tingle.) 8vo, 

Le Chatelier's High-temperature Measurements. (Boudouard — Burgess.) i2mo, 
Lob's Electrochemistry of Organic Compounds. (Lorenz.) 8vo, 

* Lyons's Treatise on Electromagnetic Phenomena. Vols. I. and II. 8vo, each, 

* Michie's Elements of Wave Motion Relating to Sound and Light 8vo, 

Niaudet's Elementary Treatise on Electric Batteries. (Fishback.) i2mo, 

* Parshall and Hobart's Electric Machine Design 4to, half morocco, 

* Rosenberg's Electrical Engineering. (Haldane Gee — Kinzbrunner.). . .8vo, 

Ryan, Norris, and Hoxie's Electrical Machinery. Vol. 1 8vo, 

Thurston's Stationary Steam-engines 8vo, 

* Tillman's Elementary Lessons in Heat 8vo, 

Tory and Pitcher's Manual of Laboratory Physics Small 8vo, 

Ulke's Modern Electrolytic Copper Refining 8vo, 



LAW. 

* Davis's Elements of Law 8vo, 

* Treatise on the Military Law of United States 8vo, 

* Sheep, 

Manual for Courts-martial , i6mo, morocco, 

Wait's Engineering and Architectural Jurisprudence c 8vo, 

Sheep, 
Law of Operations Preliminary to Construction in Engineering and Archi- 
tecture 8vo 7 

Sheep, 

Law of Contracts 8vo, 

Winthrop's Abridgment of Military Law i2mo, 



MANUFACTURES. 

Bernadou's Smokeless Powder — Nltro-celluiosc and Theory of the Cellulose 

Molecule i2mo, 

Bolland's Iron Founder i2mo, 

The Iron Founder," Supplement i2mo, 

Encyclopedia of Founding and Dictionary of Foundry Terms Used in the 

Practice of Moulding i2mo, 

Claassen's Beet-sugar Manufacture. (Hall and Rolfe.) 8vo. 

* Eckel's Cements, Limes, and Plasters 8vo, 

Eissler's Modern High Explosives. . . . .• 8vo, 

Effront's Enzymes and their Applications. (Prescott.) 8vo, 

Fitzgerald's Boston Machinist . i2mo, 

Ford's Boiler Making for Boiler Makers i8mo, 

Hopkin's Oil-chemists' Handbook 8vo, 

Keep's Cast Iron , 8vo, 

11 



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Xeach's The Inspection and Analysis of Food with Special Reference to State 
Control Large 8vo, 

* McKay and Larsen's Principles and Practice of Butter-making 8vo, 

JMatthews's The Textile Fibres 8vo, 

Metcalf's Steel. A Manual for Steel-users i2mo, 

Metcalfe's Cost of Manufactures — And the Administration of Workshops. 8vo, 

Meyer's Modern Locomotive Construction 4to, 

Morse's Calculations used in Cane-sugar Factories i6mo, morocco, 

* Reisig's Guide to Piece-dyeing 8vo, 25 

Rice's Concrete-block Manufacture - 8vo, 

Sabin's Industrial and Artistic Technology of Paints and Varnish 8vo, 

Smith's Press-working of Metals ' 8vo, 

Spalding's Hydraulic Cement i2mo, 

Spencer's Handbook for Chemists of Beet-sugar Houses i6mo, morocco, 

Handbook for Cane Sugar Manufacturers T.6mo, morocco, 

Taylor and Thompson's Treatise on Concrete, Plain and Reinforced 8vo, 

Thurston's Manual of Steam-boilers, their Designs, Construction and Opera- 
tion 8vo, 

* Walke's Lectures on Explosives 8vo, 

Ware's Beet-sugar Manufacture and Refining Small 8vo, 

Weaver's Military Explosives .8vo, 

West's American Foundry Practice i2mo, 

Moulder's Text-book i2mo, 

Wolff's Windmill as a Prime Mover 8vo, 

Wood's Rustless Coatings: Corrosion and Electrolysis of Iron and Steel. .8vo, 



MATHEMATICS. 

Baker's Elliptic Functions 8vo, 

* Bass's Elements of Differential Calculus i2mo, 

Briggs's Elements of Plane Analytic Geometry i2mo, 

Compton's Manual of Logarithmic Computations 12 mo, 

Davis's Introduction to the Logic of Algebra 8vo, 

* Dickson's College Algebra Large i2mo, 

* Introduction to the Theory of Algebraic Equations Large i2mo, 

Emch's Introduction to Projective Geometry and its Applications 8vo, 

Halsted's Elements of Geometry 8vo, 

Elementary Synthetic Geometry 8vo, 

Rational Geometry i2mo, 

* Johnson's (J. B.) Three-place Logarithmic Tables: Vest-pocket size. paper, 

100 copies for 

* Mounted on heavy cardboard, 8X 10 inches, 

10 copies for 
Johnson's (W W.) Elementary Treatise on Differential Calculus. .Small 8vo, 

Elementary Treatise on the Integral Calculus Small 8vo, 

Johnson's (W. W.) Curve Tracing in Cartesian Co-ordinates i2mo, 

Johnson's (W W.) Treatise on Ordinary and Partial Differential Equations. 

Small 8vo, 
Johnson's (W, W.) Theory of Errors and the Method of Least Squares. i2mo, 

* Johnson's (W W.) Theoretical Mechanics i2mo, 

Laplace's Philosophical Essay on Probabilities. (Truscott and Emory.) . i2mo, 

* Ludlow and Bass. Elements of Trigonometry and Logarithmic and Other 

Tables . . 8vo r 

Trigonometry and Tables published separately, Each, 

"* Ludlow's Logarithmic and Trigonometric Tables 8vo, 

^Manning's Irrational Numbers and their Representation by Sequences and Series 

T2mo 1 25 
12 



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Mathematical Monographs. Edited by Mansfield Merriman and Robert 

S. Woodward Octavo, each i oo 

No. i. History of Modern Mathematics, by David Eugene Smith. 
No. 2. Synthetic Projective Geometry, by George Bruce Halsted. 
No. 3. Determinants, by Laenas Gifford Weld. No. 4. Hyper- 
bolic Functions, by James McMahon. No. 5. Harmonic Func- 
tions, by William E. Byerly. No, 6. Grassmann's Space Analysis, 
by Edward W„ Hyde. No. 7. Probability and Theory of Errors, 
by Robert S. Woodward. No. 8. Vector Analysis and Quaternions, 
ty Alexander Macfarlane. No. 9. Differential Equations, by 
William Woolsey Johnson. No. 10. The Solution of Equations, 
by Mansfield Merriman. No. ri. Functions of a Complex Variable, 
by Thomas S. Fiske. 

-Maurer's Technical Mechanics. c . 8vo, 4 00 

Merriman v s Method of Least Squares 8vo, 2 00 

Hice and Johnson's Elementary Treatise on the Differential Calculus. . Sm. 8vo, 3 00 

Differential and Integral Calculus. 2 vols, in one. Small 8vo, 2 50 

Wood's Elements of Co-ordinate Geometry 8vo, 2 00 

Trigonometry: Analytical, Plane? and Spherical i2mo, 1 00 



MECHANICAL ENGINEERING. 

MATERIALS OF ENGINEERING, STEAM-ENGINES AND BOILERS. 

Bacon's Forge Practice. , i2mo, 

Baldwin's Steam Heating for Buildings i2mo, 

Barr's Kinematics of Machinery 8vo, 

* Bartlett's Mechanical Drawing 8vo, 

* " " " Abridged Ed 8vo, 

Benjamin's Wrinkles and Recipes i2mo, 

Carpenter's Experimental Engineering 8vo. 

Heating and Ventilating Buildings .8vo, 

Gary's Smoke Suppression in Plants using Bituminous Coal. (In Prepara- 
tion.) 

Clerk's Gas and Oil Engine Small 8vo, 

Coolidge's Manual of Drawing 8vo, paper, 

Coolidge and Freeman's Elements of General Drafting for Mechanical En- 
gineers Oblong 410, 

Cromwell's Treatise on Toothed Gearing i2mo, 

Treatise on Belts and Pulleys i2mo, 

Durley's Kinematics of Machines 8vo, 

Flather's Dynamometers and the Measurement of Power. 1 21110, 

Rope Driving i2mo, 

Gill's Gas and Fuel Analysis for Engineers i2mo, 

Hall's Car Lubrication i2mo, 

Bering's Ready Reference Tables (Conversion Factors) i6mo, morocco, 

Button's The Gas Engine 8vo, 

Jamison's Mechanical Drawing. . 8vo, 

Jones's Machine Design: 

Part I. Kinematics of Machinery. . 8vo, 

Part II. Form, Strength, and Proportions of Parts 8vo, 

"Kent's Mechanical Engineers' Pocket-book i6mo, morocco, 

Kerr's Power and Power Transmission 8vo, 

Leonard's Machine Shop, Tools, and Methods 8vo, 

* Lorenz's Modern Refrigerating Machinery. (Pope, Haven, and Dean.) . . 8vo, 
MacCord's Kinematics; or Practical Mechanism 8vo, 

Mechanical Drawing 4to. 

Velocity Diagrams. ■_ 8vo» 

13 



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3 


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3 


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3 


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3 


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3 


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3 


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3 


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I 


00 


7 


5~ 


5 


OO* 


5 


00 


3 


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2 


5<* 



MacFarland's Standard Reduction Factors for Gases 8vo, 

Mahan's Industrial Drawing. (Thompson. ) c 8vo, 

Poole's Calorific Power of Fuels 8vo, 

Reid's Course in Mechanical Drawing 8vo, 

Text-book of Mechanical Drawing and Elementary Machine Design. 8vo, 

Richard's Compressed Air i2mo, 

Robinson's Principles of Mechanism 8vo, 

Schwamb and Merrill's Elements of Mechanism 8vo, 

Smith's (O.) Press-working of Metals 8vo, 

Smith (A. W.) and Marx's Machine Design 8vo, 

Thurston's Treatise on Friction and Lost Work in Machinery and Mill 
Work.. . o 8vo, 

Animal as a Machine and Prime Motor, and the Laws of Energetics. i2mo, 

Warren's Elements of Machine Construction and Drawing 8vo, 

Weisbach's Kinematics and the Power of Transmission. (Herrmann — 
Klein.). „ 8vo, 

Machinery of Transmission and Governors. (Herrmann — Klein.). .8vo, 

Wolff's Windmill as a Prime Mover 8vo, 

Wood's Turbines 8vo, 



MATERIALS OP ENGINEERING. 

^Bovey's Strength of Materials and Theory of Structures 8vo, 7 50 

Burr's Elasticity and Resistance of the Materials of Engineering. 6th Edition. 

Reset 8vo, 

Church's Mechanics of Engineering 8vo, 

* Greene's Structural Mechanics 8vo, 

Johnson's Materials of Construction 8vo, 

Keep's Cast Iron. 8vo, 

Lanza's Applied Mechanics 8vo, 

Martens's Handbook on Testing Materials. (Henning.) 8vo, 

Maurer's Technical Mechanics 8vo, 

Merriman's Mechanics of Materials 8vo, 

Strength of Materials i2mo, 

Metcalf's Steel. A manual for Steel-users i2mo, 

Sabin's Industrial and Artistic Technology of Paints and Varnish 8vo, 

Smith's Materials of Machines i2mo, 

Thurston's Materials of Engineering 3 vols., 8vo, 

Part II. Iron and Steel 8vo, 

Part III. A Treatise on Brasses, Bronzes, and Other Alloys and their 
Constituents 8vo, 

Text-book of the Materials of Construction 8vo, 

Wood's (De V.) Treatise on the Resistance of Materials and an Appendix on 
the Preservation of Timber 8vo, 

Elements of Analytical Mechanics 8vo , 

Wood's (M. P.) Rustless Coatings: Corrosion and Electrolysis of Iron and 

Steel. > 8vo, 4 oo> 



STEAM-ENGINES AND BOILERS. 

Berry's Temperature-entropy Diagram i2mo, 1 25 

Carnot's Reflections on the Motive Power of Heat. (Thurston.), , . ,.i2mo, 1 50 

Dawson's "Engineering" and Electric Traction Pocket-book. . . .i6mo mor., 5 00 

Ford's Boiler Making for Boiler Makers i8mo, 1 00 

Goss's Locomotive Sparks 8vo, 2 00 

Hemenway's Indicator Practice and Steam-engine Economy i2mo, 2 oa 

14 



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2 


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3 


50 


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50 


5 


oa 


2 


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3 


00 



Hutton's Mechanical Engineering of Power Plants. 8vo, 5 00 

Heat and Heat-engines 8vo 5 00 

Kent's Steam boiler Economy 8vo, 4 00 

Kneass's Practice and Theory of the Injector 8vo, 1 50 

MacCord's Slide-valves 8vo, 2 00 

Meyer's Modern Locomotive Construction 4to, 10 00 

Peabody's Manual of the Steam-engine Indicator f2mo, 1 50 

Tables of the Properties of Saturated Steam and Other Vapors 8vo, 1 00 

Thermodynamics of the Steam-engine and Other Heat-engines, 8vo, 5 00 

Valve-gears for Steam-engines 8vo, 2 50 

Peabody and Miller's Steam-boilers 8vo, 4 00 

Pray's Twenty Years with the Indicator Large 8vo, 2 50 

Pupin's Thermodynamics of Reversible Cycles in Gases and Saturated Vapors. 

(Osterberg.). . i2mo, 1 25 

Reagan's Locomotives: Simple Compound, and Electric 121110, 2 50 

Rontgen's Principles of Thermodynamics. (Du Bois.) 8vo, 5 o<d 

Sinclair's Locomotive Engine Running and Management i2mo, 2 00 

Smart's Handbook of Engineering Laboratory Practice i2mo, 2 50 

Snow's Steam-boiler Practice. 8vo, 3 00 

Spangler's Valve-gears 8vo, 2 50 

Notes on Thermodynamics i2mo, 1 00 

Spangler, Greene, and Marshall's Elements of Steam-engineering 8vo, 3 00 

Thomas's Steam-turbines 8vo, 3 50 

Thurston's Handy Tables 8vo, 1 50 

Manual of the Steam-engine 2 vols., 8vo, 10 00 

Part I. History, Structure, and Theory 8vo, 6 00 

Part II. Design, Construction, and Operation 8vo, 6 00 

Handbook of Engine and Boiler Trials, and the Use of the Indicator and 

the Prony Brake 8vo, 5 00 

Stationary Steam-engines. , 8vo, 2 50 

Steam-boiler Explosions in Theory and in Practice i2mo, 1 50 

Manual of Steam-boilers, their Designs, Construction, and Operation 8vo, 5 00 

Wehrenfenning's Analysis and Softening of Boiler Feed-water (Patterson) 8vo, 4 00 

Weisbach's Heat, Steam, and Steam-engines. (Du Bois.) 8vo, 5 00 

Whitham's Steam-engine Design 8vo, 5 00 

Wood's Thermodynamics, Heat Motors, and Refrigerating Machines. . .8vo, 4 00 



MECHANICS AND MACHINERY. 



2 50 



Barr's Kinematics of Machinery 8vo, 

* Bovey's Strength of Materials and Theory of Structures 8vo, 7 50 

Chase's The Art of Pattern-making i2mo, 2 50 

Church's Mechanics of Engineering 8vo, 6 



00 



Notes and Examples in Mechanics. . , .8vo, 2 oo 

Compton's First Lessons in Metal- working i2mo, 1 50 

Compton and De Groodt's The Speed Lathe i2mo, 1 50 

Cromwell's Treatise on Toothed Gearing i2mo, 1 50 

Treatise on Belts and Pulleys i2mo, 1 50 

Dana's Text-book of Elementary Mechanics for Colleges and Schools. . 121110, 1 50 

Dingey's Machinery Pattern Making i2mo, 2 00 

Dredge's Record of the Transportation Exhibits Building of the World's 

Columbian Exposition of 1893. , 4to half morocco, 5 00 

u Bois's Elementary Principles of Mechanics • 

Vol. I. Kinematics. . 8vo, 

Vol. II. Statics „ . 8vo, 

Mechanics of Engineering. Vol. I. Small 4to, 

Vol. II Small 4to, 

Durley's Kinematics of Machines 8vo, 

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Fitzgerald's Boston Machinist i6mo # i oo> 

Flather's Dynamometers, and the Measurement of Power 121110, 3 oo> 

Rope Driving. : i2mo, 2 00 

Goss's Locomotive Sparks 8vo, 2 oa 

* Greene's Structural Mechanics 8vo, 2 5a 

Hall's Car Lubrication i2mo, 1 00 

Holly's Art of Saw Filing i8mo, 75 

James's Kinematics of a Point and the Rational Mechanics of a Particle. 

Small 8vo, 2 oa 

* Johnson's (Wo W.) Theoretical Mechanics 121210, 3 o<> 

Johnson's (L. J.) Statics by Graphic and Algebraic Methods * . . .8vo, 2 oo> 

Jones's Machine Design: 

Part I. Kinematics of Machinery ._ .8vo, 1 50- 

Part II. Form, Strength, and Proportions of Parts. 8vo, 3 00 

Kerr's Power and Power Transmission 8vo, 2 oa 

Lanza's Applied Mechanics 8vo, 7 50 

Leonard's Machine Shop, Tools, and Methods 8vo, 4 00 

* Lorenz's Modern Refrigerating Machinery. (Pope, Haven, and Dean.).8vo, 4 00 
MacCord's Kinematics; or. Practical Mechanism. . 8vo, 5 oa 

Velocity Diagrams. , . * 8vo, 1 5a 

* Martin's Text Book on Mechanics, Vol. I, Statics i2mo, 1 25 

Maurer's Technical Mechanics 8vo, 4 oa 

Merriman's Mechanics of Materials • 8vo, 5 00 

* Elements of Mechanics i2mo, 1 00 

* Michie's Elements of Analytical Mechanics 8vo, 4 oa 

* Parshall and Hobart's Electric Machine Design 4to, half morocco, 12 5a 

Reagan's Locomotives. Simple, Compound, and Electric. . . i2mo, 2 50 

Reid's Course in Mechanical Drawing. 8vo, 2 oa 

Text-book of Mechanical Drawing and Elementary Machine Design. 8vo, 3 oa 

Richards's Compressed Air * i2mo, 1 50 

Robinson's Principles of Mechanism. 8vo, 3 00 

Ryan, Norris, and Hoxie's Electrical Machinery. Vol. I .8vo, 2 50 

Sanborn's Mechanics : Problems Large i2mo, 1 50 

Schwamb and Merrill's Elements of Mechanism. . , 8vo, 3 oo 

Sinclair's Locomotive-engine Running and Management. . . i2mo, 2 oa 

Smith's (O.) Press-working of Metals 8vo, 3 oa 

Smith's (A. W.) Materials of Machines i2mo, 1 00 

Smith (A. W.) and Marx's Machine Design. 8vo, 3 oa 

Spangler, Greene, and Marshall's Elements of Steam-engineering 8vo r 3 oa 

Thurston's Treatise on Friction and Lost Work in Machinery and Mill 

Work ; 8vo, 3 oa 

Animal as a Machine and Prime Motor, and the Laws of Energetics. i2mo, 1 oa 

Warren's Elements of Machine Construction and Drawing 8vo, 7 5a 

Weisbach's Kinematics and Power of Transmission. (Herrmann — Klein. ) . 8vo , 5 oa 

Machinery of Transmission and Governors. (Herrmann — Klein. ).8vo, 5 oo 

Wood's Elements of Analytical Mechanics 8vo, 3 oa 

Principles of Elementary Mechanics i2mo, 1 25 

Turbines. . ' 8vo, 2 5a 

The World's Columbian Exposition of 1893 , 4to, 1 oa 



METALLURGY. 

Egleston's Metallurgy of Silver, Gold, and Mercury: 

Vol. I. Silver 8vo, 7 5a 

Vol. II. Gold and Mercury. . 8vo, 7 50 

Goesel's Minerals and Metals; a Reference Book , . . . . i6mo, mor. 3 00 

** Iles's Lead-smelting. (Postage 9 cents additional.). - i2mo, 2 5a 

Keep's Cast Iron 8vo, 2 50 

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Kunhardt's Practice of Ore Dressing in Europe. 8vo, 

Le Chatelier's High-temperature Measurements. (Boudouard — Burgess. )i2mo. 

Metcalf's Steel. A Manual for Steel-users , nmo, 

Miller's Cyanide Process 12010, 

Minet's Production of Aluminum and its Industrial Use. (Waldo.). . . . i2mo, 

Robine and Lenglen's Cyanide Industry. (Le Clerc). . „ 8vo, 

Smith's Materials of Machines nmo, 

Thurston's Materials of Engineering. In Three Parts 8vo, 

Part II. Iron and Steel 8vo, 

Part III. A Treatise on Brasses, Bronzes, and Other Alloys and their 

Constituents 8vo, 

Ulke's Modern Electrolytic Copper Refining 8vo, 



MINERALOGY. 

Barringer's Description of Minerals of Commercial Value. Oblong, morocco, 

Boyd's Resources of Southwest Virginia .8vo, 

Map of Southwest Virignia Pocket-book form. 

Brush's Manual of Determinative Mineralogy. (Penfield.) 8vo, 

Chester's Catalogue of Minerals 8vo, paper, 

Cloth, 

Dictionary of the Names of Minerals 8vo 

Dana's System of Mineralogy Large 8vo, half leather 

First Appendix to Dana's New " System of Mineralogy." Large 8vo, 

Text-book of Mineralogy 8vo, 

Minerals and How to Study Them i2mo. 

Catalogue of American Localities of Minerals. . . Large 8vo, 

Manual of Mineralogy and Petrography i2mo- 

Douglas's Untechnical Addresses on Technical Subjects i2mo, 

Eakle's Mineral Tables * 8vo, 

Egleston's Catalogue of Minerals and Synonyms 8vo, 

Goesel's Minerals and Metals: A Reference Book i6mo,mor.. 

Groth's Introduction to Chemical Crystallography (Marshall) 12 mo, 

Hussak's The Determination of Rock-forming Minerals. ( Smith.). Small 8 vo> 
Merrill's Non-metallic Minerals* Their Occurrence and Lses 8vo, 

* Penfield's Notes on Determinative Mineralogy and Record of Mineral Tests. 

8vo, paper, 50 
Rosenbusch's Microscopical Physiography of the Rock-making Minerals. 

(Iddings.) 8vo, 5 oa 

* Tillman's Text-book of Important Minerals and Rocks 8vo, 2 00 



MINING. 

Beard's Ventilation of Mines i2mo, 

Boyd's Resources of Southwest Virginia 8vo, 

Map of Southwest Virginia. , Pocket-book form, 

Douglas's Untechnical Addresses on Technical Subjects i2mo, 

* Drinker's Tunneling, Explosive Compounds > and Rock Drills. .4to,hf. rnor., 

Eissler's Modern High Explosives. ... 8~-> 

Goesel's Minerals and Metals • A Reference Book i6mo, mor. 

Goodyear's Coal-mines of the V, estern Coast of the United States i2mo, 

Ihlseng's Manual of Mining 8vo, 

** Iles's Lead-smelting. (Postage ox. additional.). .. i2mo, 

Kunhardt's Practice of Ore Dressing in Europe. . . 8vo, 

Miller's Cyanide Process i2mo, 

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'O'Driscoll's Notes on the Treatment of Gold Ores. 8vo, 

Robine and Lenglen's Cyanide Industry. (Le Clerc). . 8vo, 

'* Walke's Lectures on Explosives 8vo, 

Weaver's Military Explosives 8vo, 

Wilson's Cyanide Processes i2mo, 

Chlorination Process. i2mo, 

Hydraulic and Placer Mining i2mo, 

Treatise on Practical and Theoretical Mine Ventilation T2mo, 



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SANITARY SCIENCE. 



Bashore's Sanitation of a Country House i2mo 

:: Outlines of Practical Sanitation. . i2mo 

Folwell's Sewerage. (Designing, Construction, and Maintenance.) 8vo 

Water-supply Engineering 8vo 

Fowler's Sewage Works Analyses i2mo 

Fuertes's Water and Public Health i2mo 

Water-filtration Works i2mo 

Gerhard's Guide to Sanitary House-inspection i6mo 

Goodrich's Economic Disposal of Town's Refuse. . .'...< Demy 8vo 

Hazen's Filtration of Public Water-supplies 8vo 

Leach's The Inspection and Analysis of Food with Special Reference to State 

Control 8vo 

Mason's Water-supply. (Considered principally from a Sanitary Standpoint) 8vo 

Examination of Water. (Chemical and Bacteriological.) i2mo 

' Ogden's Sewer Design i2mo 

Prescott and Winslow's Elements of Water Bacteriology, with Special Refer- 
ence to Sanitary Water Analysis i2mo 

* Price's Handbook on Sanitation i2mo 

Richards's Cost of Food. A Study in Dietaries i2mo 

Costiof Living as Modified by Sanitary Science i2mo 

Cost of Shelter i2mo 

Richards and Woodman's Air, Water, and Food from a Sanitary Stand- 
point -'. 8vo 

* Richards and Williams's The Dietary Computer 8vo 

Rideal's Sewage and Bacterial Purification of Sewage 8vo 

Turneaure and Russell's Public Water-supplies 8vo 

Von Behring's Suppression of Tuberculosis. (Bolduan.) i2mo 

Whipple's Microscopy of Drinking-water Svo 

Winton's Microscopy of Vegetable Foods 8vo 

WoodhulPs Notes on Military Hygiene i6mo 

* Personal Hygiene i2mo 



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MISCELLANEOUS. 



De Fursac's Manual of Psychiatry. (Rosanoff and Collins.). . . .Large i2mo, 

Ehrlich's Collected Studies on Immunity ( Bolduan) 8vo, 

Emmons's Geological Guide-book of the Rocky Mountain Excursion of the 

International Congress of Geologists Large Svo, 

lerrel's Popular Treatise on the Winds 8vo- 

Haines's American Railway Management i2mo, 

Mott's Fallacy of the Present Theory of Sound i6mo, 

Ricketts's History of Rensselaer Polytechnic Institute, 1824-1804. .Small 8vo, 

Rostoski's Serum Diagnosis. (Bolduan.) i2mo , 

Rotherham's Emphasized New Testament .... . .Large 8vo, 

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Steel's Treatise on the Diseases of the Dog 8vo, 3 50 

The World's Columbian Lxposition of 1893 4to, 1 00 

Von Behring's Suppression of Tuberculosis. (Bolduan.) 12010, 1 00 

Winslow's Elements of Applied Microscopy i2mo, 1 50 

Worcester and Atkinson. Small Hospitals, Establishment and Maintenance; 

Suggestions for Hospital Architecture: Plans for Small Hospital. 1 2mo, 1 25 



HEBREW AND CHALDEE TEXT-BOOKS. 

Green's Elementary Hebrew Grammar i2mo, 1 25 

Hebrew Chrestomathy 8vo, 2 00 

Gesenius's Hebrew and Chaldee Lexicon to the Old Testament Scriptures. 

(Tregelles.) Small 4to, half morocco. 5 00 

Letteris's Hebrew Bible 8vo, 2 25 

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JAN 7 '907 



